Langevin dynamic for the 2D Yang-Mills measure
aa r X i v : . [ m a t h . P R ] J un Langevin dynamic for the 2D Yang–Mills measure
June 11, 2020
Ajay Chandra , Ilya Chevyrev , Martin Hairer , and Hao Shen Imperial College LondonEmail: [email protected], [email protected] University of Oxford, Email: [email protected] University of Wisconsin-Madison, Email: [email protected]
Abstract
We define a natural state space and Markov process associated to the stochasticYang–Mills heat flow in two dimensions.To accomplish this we first introduce a space of distributional connections forwhich holonomies along sufficiently regular curves (Wilson loop observables) andthe action of an associated group of gauge transformations are both well-definedand satisfy good continuity properties. The desired state space is obtained as thecorresponding space of orbits under this group action and is shown to be a Polishspace when equipped with a natural Hausdorff metric.To construct the Markov process we show that the stochastic Yang–Mills heatflow takes values in our space of connections and use the “DeTurck trick” ofintroducing a time dependent gauge transformation to show invariance, in law, ofthe solution under gauge transformations.Our main tool for solving for the Yang–Mills heat flow is the theory of regularitystructures and along the way we also develop a “basis-free” framework for applyingthe theory of regularity structures in the context of vector-valued noise – thisprovides a conceptual framework for interpreting several previous constructionsand we expect this framework to be of independent interest.
Contents ntroduction -forms . . . . . . . . . . . . . . . . . . . . . . . . . 263.4 Gauge transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.5 Holonomies and recovering gauge transformations . . . . . . . . . . . . . 323.6 The orbit space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Υ . . . . . . . . . . . . . . . . . 71 d = 2 . . . . . 84 A Symbolic index 116
The purpose of this paper and the companion article [CCHS] is to study the Langevindynamic associated to the Euclidean Yang–Mills (YM) measure. Formally, the YMmeasure is written d µ ym ( A ) = Z − exp [ − S ym ( A ) ] d A , (1.1)where d A is a formal Lebesgue measure on the space of connections of a principal G -bundle P → M , G is a compact Lie group, and Z is a normalisation constant.The YM action is given by S ym ( A ) def = Z M | F A ( x ) | d x , (1.2)where F A is the curvature -form of A , the norm | F A | is given by an Ad -invariantinner product on the Lie algebra g of G , and d x is a Riemannian volume measure ntroduction on the space-time manifold M . The YM measure plays a fundamental role inhigh energy physics, constituting one of the components of the Standard Model,and its rigorous construction largely remains open, see [JW06, Cha19] and thereferences therein. The action S ym in addition plays a significant role in geometry,see e.g. [AB83, DK90].For the rest of our discussion we will take M = T d , the d -dimensional torusequipped with a Euclidean inner product and normalised Haar measure, and theprincipal bundle P to be trivial. In particular, we will identify the space of connec-tions on P with g -valued -forms on T d (implicitly fixing a global section). Theresults of this paper almost exclusively focus on the case d = 2 , and the case d = 3 is studied in [CCHS].A postulate of gauge theory is that all physically relevant quantities should beinvariant under the action of the gauge group, which consists of the automorphismsof the principal bundle P . In our setting, the gauge group can be identified withmaps g ∈ G ∞ = C ∞ ( T d , G ) , and the corresponding action on connections is givenby A A g def = gAg − − (d g ) g − . (1.3)Equivalently, the -form A g represents the same connection as A but in a newcoordinate system (i.e. global section) determined by g . The physically relevantobject is therefore not the connection A itself, but its orbit [ A ] under the action(1.3).In addition to the challenge of rigorously interpreting (1.1) due to the infinite-dimensionality of the space of connections, gauge invariance poses an additionaldifficulty that is not encountered in theories such as the Φ p models. Indeed, since S ym is invariant under the action of the infinite-dimensional gauge group G ∞ (asit should be to represent a physically relevant theory), the interpretation of (1.1)as a probability measure on the space of connections runs into the problem ofthe impossibility of constructing a measure that is “uniform” on each gauge orbit.Instead, one would like to quotient out the action of the gauge group and build themeasure on the space of gauge orbits, but this introduces a new difficulty in that itis even less clear what the reference “Lebesgue measure” means in this case.A natural approach to study the YM measure is to consider the Langevindynamic associated with the action S ym . Indeed, this dynamic is expected to benaturally gauge covariant and one can aim to use techniques from PDE theory tounderstand its behaviour. Denoting by d A the covariant derivative associated with A and by d ∗ A its adjoint, the equation governing the Langevin dynamic is formallygiven by ∂ t A = − d ∗ A F A + ξ . (1.4)In coordinates this reads, for i = 1 , . . . , d and with summation over j implicit, ∂ t A i = ξ i + ∆ A i − ∂ ji A j + [ A j , ∂ j A i − ∂ i A j + [ A j , A i ]] + [ ∂ j A j , A i ] , (1.5) Implicit summation over repeated indices will be in place throughout the paper with departuresfrom this convention explicitly specified. ntroduction where ξ , . . . , ξ d are independent g -valued space-time white noises on R × T d with covariance induced by an ad -invariant scalar product on g . (We fix such ascalar product for the remainder of the discussion.) Equation (1.4) was the originalmotivation of Parisi–Wu [PW81] in their introduction of stochastic quantisation.This field has recently received renewed interest due to a development of tools ableto study singular SPDEs [Hai14, GIP15], and has proven fruitful in the study and analternative construction of the scalar Φ quantum field theories [MW17b, MW17a,AK17, MW18, GH18] (see also [BG18] for a related construction).A very basic issue with (1.4) is the lack of ellipticity of the term d ∗ A F A , whichis a reflection of the invariance of the action S ym under the gauge group. A well-known solution to this problem is to realise that if we take any sufficiently regularfunctional A H ( A ) ∈ C ∞ ( T , g ) and consider instead of (1.4) the equation ∂ t A = − d ∗ A F A + d A H ( A ) + ξ , (1.6)then, at least formally, solutions to (1.6) are gauge equivalent to those of (1.4) inthe sense that there exists a time-dependent gauge transformation mapping one intothe other one, at least in law. This is due to the fact that the tangent space of thegauge orbit at A (ignoring issues of regularity / topology for the moment) is givenby terms of the form d A ω , where ω is an arbitrary g -valued -form.A convenient choice of H is given by H ( A ) = − d ∗ A which yields the so-calledDeTurck–Zwanziger term [Zwa81, DeT83] − d A d ∗ A = d x i ( ∂ i + [ A i , · ]) ∂ j A j . This allows to cancel out the term ∂ ji A j appearing in (1.5) and thus renders theequation parabolic, while still keeping the solution to the modified equation gauge-equivalent to the original one. We note that the idea to use this modified equationto study properties of the heat flow has proven a useful tool in geometric analy-sis [DeT83, Don85, CG13] and has appeared in works on stochastic quantisation inthe physics literature [Zwa81, BHST87, DH87].With this discussion in mind, the equation we focus on, also referred to in thesequel as the stochastic Yang–Mills (SYM) equation, is given in coordinates by ∂ t A i = ∆ A i + ξ i + [ A j , ∂ j A i − ∂ i A j + [ A j , A i ] ] . (1.7)Our goal is to show the existence of a natural space of gauge orbits such that(appropriately renormalised) solutions to (1.7) define a canonical Markov processon this space. In addition, one desires a class of gauge invariant observablesto be defined on this orbit space which is sufficiently rich to separate points; apopular class is that of Wilson loop observables (another being the lasso variablesof Gross [Gro85]), which are defined in terms of holonomies of the connection and avariant of which is known to separate the gauge orbits in the smooth setting [Sen92].One of the difficulties in carrying out this task is that any reasonable definition forthe state space should be supported on gauge orbits of distributional connections, ntroduction and it is a priori not clear how to define holonomies (or other gauge-invariantobservables) for such connections. In fact, it is not even clear how to carry out theconstruction to ensure that the orbits form a reasonable (e.g. Polish) space, giventhat the quotient of a Polish space by the action of a Polish group will typicallyyield a highly pathological object from a measure-theoretical perspective. (Thinkof even simple cases like the quotient of L ([ , ]) by the action of H ([ , ]) givenby ( x, g ) x + ιg with ι : H → L the canonical inclusion map or the quotientof the torus T by the action of ( R , + ) given by an irrational rotation.)We now describe our main results on an informal level, postponing a preciseformulation to Section 2, and mention connections with the existing literature andseveral open problems. Our first contribution is to identify a natural space of distributional connections Ω α , which can be seen as a refined analogue of the classical Hölder–Besov spaces,along with an associated gauge group. An important feature of this space is thatholonomies along all sufficiently regular curves (and thus Wilson loops and theirvariants) are canonically defined for each connection in Ω α and are continuousfunctions of the connection and curve. In addition, the associated space of gaugeorbits is a Polish space and thus well-behaved from the viewpoint of probabilitytheory. A byproduct of the construction of Ω α is a parametrisation-independentway of measuring the regularity of a curve which relates to α -HÃűlder regularcurves with α > in a way that is strongly reminiscent of how p -variation relatesto HÃűlder regularity for α ≤ .In turn, we show that the SPDE (1.7) can naturally be solved in the space Ω α through mollifier approximations. More precisely, we show that for any mollifier χ ε at scale ε ∈ ( , ] and C ∈ L G ( g , g ) (where L G ( g , g ) consists of all linearoperators from g to itself which commute with Ad g for any g ∈ G ), the solutionsto the renormalised SPDE ∂ t A i = ∆ A i + χ ε ∗ ξ i + CA i + [ A j , ∂ j A i − ∂ i A j + [ A j , A i ] ] (1.8)converge as Ω α -valued processes as ε → (with a possibility of finite-time blow-up). Observe that the addition of the mass term in (1.8) (as well as the choice ofmollification with respect to a fixed coordinate system) breaks gauge-covariancefor any ε > . Our final result is that gauge-covariance can be restored in the ε → limit. Namely, we show that for each non-anticipative mollifier χ , thereexists a unique choice for C (depending on χ ) such that in the limit ε → , the lawof the gauge orbit [ A ( t )] is independent of χ and depends only on the gauge orbit [ A ( )] of the initial condition. This provides the construction of the aforementionedcanonical Markov process associated to (1.7) on the space of gauge orbits.We mention that a large part of the solution theory for (1.7) is now automatic andfollows from the theory of regularity structures [Hai14, BHZ19, CH16, BCCH17].In particular, these works guarantee that a suitable renormalisation procedure yields ntroduction convergence of the solutions inside some Hölder–Besov space. The points whichare not automatic are that the limiting solution indeed takes values in the space Ω α , that it is gauge invariant, and that no diverging counterterms are required forthe convergence of (1.8). One contribution of this article is to adapt the algebraicframework of regularity structures developed in [BHZ19, BCCH17] to address thelatter point. Precisely, we give a natural renormalisation procedure for SPDEs withvector-valued noise and solution of the form ( ∂ t − L t ) A t = F t ( A , ξ ) , t ∈ L + . (1.9)Here ( L t ) t ∈ L + are differential operators, A and ξ represent the jet of ( A t ) t ∈ L + and ( ξ l ) l ∈ L − which take values in vector spaces ( W t ) t ∈ L + and ( W t ) t ∈ L − respectively,and the nonlinearities ( F t ) t ∈ L + are smooth and local. We give a systematic wayto build a regularity structure associated to (1.9) and to derive the renormalisedequation without ever choosing a basis of the spaces W t . Example 1.1
In addition to (1.7) , an equation of interest which fits into this frame-work comes from the Langevin dynamic of the Yang–Mills–Higgs Lagrangian Z T d (cid:16) | F A | + | d A Φ | − m | Φ | + 14 | Φ | (cid:17) d x , (1.10) where A is a -form taking values in a Lie sub-algebra g of the anti-Hermitianoperators on C N , and Φ is a C N -valued function. The associated SPDE (againwith DeTurck term) reads ∂ t A = − d ∗ A F A − d A d ∗ A A + B ( Φ ⊗ d A Φ ) + ξ A , ∂ t Φ = − d ∗ A d A Φ + (d ∗ A A ) Φ − Φ | Φ | − m Φ + ξ Φ , (1.11) where B : C N ⊗ C N → g is the R -linear map that satisfies h h, B ( x ⊗ y ) i g =2 Re h hx, y i C N for all h ∈ g .One of the consequences of our framework is that the renormalisation counter-terms of (1.11) can all be constructed from iterated applications of B , the Liebracket [ · , · ] g , and the product ( A, Φ ) A Φ . There have been several earlier works on the construction of an orbit space. Mitter–Viallet [MV81] showed that the space of gauge orbits modelled on H k for k > d + 1 is a smooth Hilbert manifold. More recently, Gross [Gro17] has made progress onthe analogue in H / in dimension d = 3 .An alternative (but related) route to give meaning to the YM measure is todirectly define a stochastic process indexed by a class of gauge invariant observ-ables (e.g. Wilson loops). This approach was undertaken in earlier works on the2D YM measure [Dri89, Sen97, Lév03, Lév06] which have successfully given ex-plicit representations of the measure for general compact manifolds and principal ntroduction bundles. It is not clear, however, how to extract from these works a space of gaugeorbits with a well-defined probability measure, which is somewhat closer to thephysical interpretation of the measure. (This is a kind of non-linear analogue toKolmogorov’s standard question of finding a probability measure on a space of“sufficiently regular” functions that matches a given consistent collection of n -pointdistributions.) In addition, this setting is ill-suited for the study of the Langevindynamic since it is far from clear how to interpret a realisation of such a stochasticprocess as the initial condition for a PDE.A partial answer was obtained in [Che19] where it was shown that a gauge-fixed version of the YM measure (for a simply-connected structure group G ) canbe constructed in a Banach space of distributional connections which could serveas the space of initial conditions of the PDE (1.7). Section 3 of this paper extendspart of this earlier work by providing a strong generalisation of the spaces usedtherein (e.g. supporting holonomies along all sufficiently regular paths, while onlyaxis-parallel paths are handled in [Che19]) and constructing an associated canonicalspace of gauge orbits.Another closely related work was recently carried out in [She18]. It was shownthere that the lattice gauge covariant Langevin dynamic of the scalar Higgs model(the Lagrangian of which is given by (1.10) without the | Φ | term and with an abelianLie algebra) in d = 2 can be appropriately modified by a DeTurck–Zwanziger termand renormalised to yield local-in-time solutions in the continuum limit. The massrenormalisation term CA i as in (1.8) is absent in [She18] due to the fact that thelattice gauge theory preserves the exact gauge symmetry, while a divergent massrenormalisation for the Higgs field Φ is still needed but preserves gauge invariance.In addition, convergence of a natural class of gauge-invariant observables wasshown over short time intervals; but there was no description for the orbit space. It is natural to conjecture that the Markov process constructed in this paper possessesa unique invariant measure, for which the associated stochastic process indexed byWilson loops agrees with the YM measure constructed in [Sen97, Lév03, Lév06].Such a result would be one of the few known rigorous connections between the YMmeasure and the YM energy functional (1.2) (another connection is made in [LN06]through a large deviations principle). A possible approach would be to show thatthe gauge-covariant lattice dynamic for the discrete YM measure converges to thesolution to the SYM equation (1.7) identified in this paper. Combined with a gaugefixing procedure as in [Che19] and an argument of Bourgain [Bou94] along the linesof [HM18a], this convergence would prove the result (as well as strong regularityproperties of the YM measure obtained from the description of the orbit spacein this paper). The main difficulty to overcome is the lack of general stochasticestimates for the lattice which are available in the continuum thanks to [CH16].Our results do not exclude finite-time blow-up of solutions to SYM (1.7), noteven in the quotient space. (Since gauge orbits are unbounded, non-explosion ofsolutions to (1.7) is a stronger property than non-explosion of the Markov process on ntroduction gauge orbits constructed in this article). It would be of interest to determine whetherthe solution to SYM survives almost surely for all time for any initial condition.The weaker case of the Markov process would be handled by the above conjecturecombined with the strong Feller property [HM18b] and irreducibility [HS19] whichboth hold in this case. The analogous result is known for the Φ d SPDE in d =2 , [MW18]. Long-time existence of the deterministic YM heat flow in d = 2 , is also known [Rad92, CG13], but it is not clear how to adapt these methods to thestochastic setting.It is also unclear how to extend the results of this paper to the 3D setting.In [CCHS] we analyse the SPDE (1.7) for d = 3 and show a form of gauge-covariance in law, which formally should give rise to a Markov process on the orbitspace. However, it is unclear how to construct the orbit space, which is closelylinked to the fact that Wilson loop observables become singular in d = 3 . We givefurther details therein. The paper is organised as follows. In Section 2, we give a precise formulation ofour main results concerning the SPDE (1.7) and the associated Markov process ongauge orbits. In Section 3 we provide a detailed study of the space of distributional -forms Ω α used in the construction of the state space of the Markov process. InSection 4 we study the stochastic heat equation as an Ω α -valued process.In Section 5 we give a canonical, basis-free framework for constructing reg-ularity structures associated to SPDEs with vector-valued noise. Moreover, wegeneralise the main results of [BCCH17] on formulae for renormalisation countert-erms in the scalar setting and obtain analogous vectorial formulae. We expect thisframework to be useful in for a variety of systems of SPDE whose natural formu-lation involve vector-valued noise – in the context of (1.7) this framework allowsus to directly obtain expressions for renormalisation counterterms in terms of Liebrackets and to use symmetry arguments coming from the Ad -invariance of thenoises.In Section 6 we prove local well-posedness of the SPDE (1.7), and in Section 7we show that gauge covariance holds in law for a specific choice of renormalisationprocedure which allows us to construct the canonical Markov process on gaugeorbits. We collect some notation and definitions used throughout the paper. We denote by R + the interval [ , ∞ ) and we identify the torus T with the set [ − , ) . We equip T with the geodesic distance, which, by an abuse of notation, we denote | x − y | ,and R × T with the parabolic distance | ( t, x ) − ( s, y ) | = p | t − s | + | x − y | .A mollifier χ is a smooth function on space-time R × R (or just space R ) withsupport in the ball { z | | z | < } such that R χ = 1 . We will assume that any space- ntroduction time mollifier χ we use satisfies χ ( t, x , x ) = χ ( t, − x , x ) = χ ( t, x , − x ) . Aspace-time mollifier is called non-anticipative if it has support in the set { ( t, x ) | t ≥ } .Consider a separable Banach space ( E, | · | ) . For α ∈ [ , ] and a metric space ( F, d ) , we denote by C α -Höl ( F, E ) the set of all functions f : F → E such that | f | α -Höl def = sup x,y | f ( x ) − f ( y ) | d ( x, y ) α < ∞ ,where the supremum is over all distinct x, y ∈ F . We further denote by C α ( F, E ) the space of all functions f : F → E such that | f | C α def = | f | ∞ + | f | α -Höl < ∞ ,where | f | ∞ def = sup x ∈ F | f ( x ) | . For α > , we define C α ( T , E ) (resp. C α ( R × T , E ) ) to be the space of k -times differentiable functions (resp. functions that are k -times differentiable in t and k -times differentiable in x with k + k ≤ k ),where k def = ⌈ α ⌉ − , with ( α − k ) -Hölder continuous k -th derivatives.For α < , let r def = −⌈ α − ⌉ and B r denote the set of all smooth functions ψ ∈ C ∞ ( T ) with | ψ | C r ≤ and support in the ball | z | < . Let ( C α ( T , E ) , | · | C α ) denote the space of distributions ξ ∈ D ′ ( T , E ) for which | ξ | C α def = sup λ ∈ ( , ] sup ψ ∈B r sup x ∈ T |h ξ, ψ λx i| λ α < ∞ ,where ψ λx ( y ) def = ε − d ψ ( ε − ( y − x )) . For α = 0 , we define C to simply be L ∞ ( T , E ) ,and use C ( T , E ) to denote the space of continuous functions, both spaces beingequipped with the L ∞ norm. For any α ∈ R , we denote by C ,α the closure ofsmooth functions in C α . We drop E from the notation and write simply C ( T ) , C α ( T ) , etc. whenever E = R .For a space B of E -valued functions (or distributions) on T , we denote by Ω B the space of E -valued -forms A = P i =1 A i d x i where A , A ∈ B . If B isequipped with a (semi)norm | · | B , we define | A | Ω B def = X i =1 | A i | B . When B is of the form C ( T , E ) , C α ( T , E ) , etc., we write simply Ω C , Ω C α , etc.for Ω B .Given two real vector spaces V and W we write L ( V, W ) for the set of alllinear operators from V to be W . If V is equipped with a topology, we write V ∗ for its topological dual, and otherwise we write V ∗ for its algebraic dual. Asmentioned in the introduction, we also write L G ( g , g ) = { C ∈ L ( g , g ) : C Ad g = Ad g C for all g ∈ G } . This assumption is for convenience, so that some constants in our renormalisation calculationvanish, but is not strictly necessary. ain results
We would like to thank Andris Gerasimovičs for many discussions regarding the derivationof the renormalised equation in Sections 6 and 7. MH gratefully acknowledges supportby the Royal Society through a research professorship. IC is funded by a Junior ResearchFellowship of St John’s College, Oxford. HS gratefully acknowledges support by NSFDMS-1712684 / DMS-1909525 and DMS-1954091. AC gratefully acknowledges financialsupport from the Leverhulme Trust via an Early Career Fellowship, ECF-2017-226.
In this section, we give a precise formulation of the main results described in theintroduction.
Our first result concerns the state space of the Markov process. We collect the mainfeatures of this space in the following theorem along with precise references, andrefer the reader to Section 3 for a detailed study.
Theorem 2.1
For each α ∈ ( , ) , there exists a Banach space Ω α of distributional g -valued -forms on T with the following properties.(i) For each A ∈ Ω α and γ ∈ C ,β ([ , ] , T ) with β ∈ ( α − , ] , the holonomy hol ( A, γ ) ∈ G is well-defined and, on bounded balls of Ω α × C ,β ([ , ] , T ) ,is a Hölder continuous function of ( A, γ ) with distances between γ ’s measuredin the supremum metric. In particular, Wilson loop observables are well-defined on Ω α . (See Theorem 3.18 and Proposition 3.21 combined withYoung ODE theory [Lyo94, FH14].)(ii) There are canonical embeddings with the classical Hölder–Besov spaces Ω C ,α/ ֒ → Ω α ֒ → Ω C ,α − . (See Section 3.3.)(iii) Let G ,α denote the closure of smooth functions in C α -Höl ( T , G ) . Thenthere is a continuous group action of G ,α on Ω α such that O α def = Ω α / G ,α equipped with the quotient topology is a Polish space. (See Corollary 3.36and Theorem 3.45.)(iv) Gauge orbits in O α are uniquely determined by conjugacy classes of holonomiesalong loops. (See Proposition 3.35.) Remark 2.2
Analogous spaces could be defined on any manifold, but it is not clearwhether higher dimensional versions are useful for the study of the stochastic YMequation.
Remark 2.3
Since hol ( A, γ ) is independent of the parametrisation of γ , the “right”way of measuring its regularity should also be parametrisation-independent, which ain results is not the case of C ,β . This is done in Definition 3.16 which might be of independentinterest.We now turn to the results on the SPDE. Let us fix α ∈ ( , ) and η ∈ ( − , α − ] .We denote Ω α,T def = C ([ , T ) , Ω α ) . Furthermore, let Y = Ω C η ∪ { } equippedwith the topology whose basis sets are the balls of Ω C η and sets of the form { A ∈ Y | | A | C η > N } for N ≥ , where we use the convention | | C η def = ∞ .For A ∈ C ( R + , Y ) and L ∈ ( , ∞ ] , let T L ( A ) def = inf { t ≥ | | A ( t ) | C η ≥ L } . We set Ω sol def = n A ∈ C ( R + , Y ) | A ↾ [ ,T ∞ ( A )) ∈ Ω α,T ∞ ( A ) , A ↾ [ T ∞ ( A ) , ∞ ) ≡ o . We equip Ω sol with the metric d ( · , · ) def = P ∞ L =1 − L d L ( · , · ) , where d L ( A, ¯ A ) def = 1 ∧ n sup t ∈ [ ,L ] | Θ L ( A )( t ) − Θ L ( ¯ A )( t ) | C η + | Θ L ( A ) − Θ L ( ¯ A ) | Ω α,L o and Θ L ( A ) def = A ( t ) t
The solution A ε converges in Ω sol in probabilityas ε → to an Ω sol -valued random variable A . Remark 2.5
Note that we could take the initial condition a ∈ Ω C η , and the analo-gous statement would hold at the expense of changing the definition of Ω α,T aboveto C (( , T ) , Ω α ) (that is, we lose continuity at t = 0 ). Remark 2.6
As one would expect, the roughest part of the solution A is alreadycaptured by the solutions Ψ to the stochastic heat equation. (In fact, one has A = Ψ + B where B belongs to in C − κ for any κ > .) Hence, fine regularityproperties of A can be inferred from those of Ψ . In particular, one could sharpenthe above result to encode time regularity of the solution A at the expense of takingsmaller values of α , cf. Theorem 4.13. ain results From our assumption that G is compact, it follows that g is reductive,namely it can be written as the direct sum of simple Lie algebras and an abelian Liealgebra. Note that if h is one of the simple components, then every C ∈ L G ( g , g ) preserves h and its restriction to h is equal to λ id h for some λ ∈ R ; indeed, h is theLie algebra of a compact Lie group (see the proof of [Kna02, Thm.4.29] for a similarstatement) and thus its complexification is also simple, and the claim follows readilyfrom Schur’s lemma. Furthermore, since these components are orthogonal underthe Ad-invariant inner product on g introduced in (1.2), each white noise ξ i alsosplits into independent noises, each valued in the abelian or a simple component.Eq. (2.1) then decouples into a system of equations each for a simple or abeliancomponent, which means that it suffices to prove Theorem 2.4 in the case of asimple Lie algebra for which we can take C ∈ R (this is the approach we take inour analysis of this SPDE). In the abelian case, (2.1) is just a linear stochastic heatequation taking values in an abelian Lie algebra with C a linear map (commutingwith Ad ) from the abelian Lie algebra to itself, for which the solution theory isstandard.We give the proof of Theorem 2.4 in Section 6. In principle a large part of theproof is by now automatic and follows from the series of results [Hai14, CH16,BHZ19, BCCH17]. Key facts which don’t follow from general principles are thatthe solution takes values in the space Ω α (but this only requires one to show that theSHE takes values in it) and more importantly that no additional renormalisation isrequired. However, if one were to directly apply the framework of [Hai14, CH16,BHZ19, BCCH17], one would have to expand the system with respect to a basisof g into a system of equations driven by d × dim( g ) independent R -valued scalar space-time white noises. The renormalised equation computed using [BCCH17]would then have to be rewritten to be taken back to the setting of vector valuednoises. In particular, verifying that the renormalisation counterterm takes the formprescribed above would be both laborious and not very illuminating. We insteadchoose to work with (2.1) intrinsically and, in Section 5, develop a framework forapplying the theory of regularity structures and the formulae of [BCCH17] directlyto equations with vector valued noise.When working with scalar noises, a labelled decorated combinatorial tree τ ,which represents some space-time process, corresponds to a one dimensional sub-space of our regularity structure. On the other hand, if our noises take values insome vector space W , then it is natural for τ to index a subspace of our regularitystructure isomorphic to a partially symmetrised tensor product of copies of W ∗ ,where the particular symmetrisation is determined by the symmetries of τ .One of our key constructions in Section 5 is a functor F · ( • ) which maps labelleddecorated combinatorial trees, which we view as objects in a category of “symmetricsets”, to these partially symmetrised tensor product spaces in the category of vectorof spaces. In other words, operations / morphisms between these trees analogousto the products and the coproducts of [BHZ19] are mapped, under this functor, to See Section 5.1 for more detail on why this is indeed natural. ain results corresponding linear maps between the vector spaces they index. This allows us toconstruct a regularity structure, with associated structure group and renormalisationgroup, without performing any basis expansions.We also show that this functor behaves well under direct sum decompositions ofthe vector spaces W , which allows us to verify that our constructions in the vectornoise setting are consistent with the regularity structure that would be obtained in thescalar setting if one performed a basis expansion. This last point allows us to transferresults from the setting of scalar noise to that of vector noise. One of our mainresults in that section is Proposition 5.65 which reformulates the renormalisationformulae of [BCCH17] in the vector noise setting. The reader may wonder why we don’t simply enforce C = 0 in (2.1) since thisis allowed in our statement. One reason is that although the limit of A ε existsfor such a choice, it would depend in general on the choice of mollifier χ . Moreimportantly, our next result shows that it is possible to counteract this by choosing C as a function of χ in such a way that not only the limit is independent of thechoice of χ , but the canonical projection of A onto O α is independent (in law!) ofthe choice of representative of the initial condition. This then allows us to use thisSPDE to construct a “nice” Markov process on the gauge orbit space O α , whichwould not be the case for any other choice of C .We first discuss the (lack of) gauge invariance of the mollified equation (2.1)from a geometric perspective. Recall that the natural state space for A is the space A of (for now smooth) connections on a principal G -bundle P (which we assumeis trivial for the purpose of this article). The space of connections is an affine spacemodelled on the vector space Ω ( T , ad( P )) , the space of -forms on T with valuesin the adjoint bundle. In what follows, we drop the references to T and ad( P ) .Recall furthermore that the covariant derivative is a map d A : Ω k → Ω k +1 withadjoint d ∗ A : Ω k +1 → Ω k . Hence, the correct geometric form of the DeTurck–Zwanziger term d A d ∗ A is really d A d ∗ Z ( A − Z ) = d A d ∗ A ( A − Z ) , where Z isthe canonical flat connection associated with the global section of P which weimplicitly chose at the very start (this choice for Z is only for convenience – anyfixed “reference” connection Z will lead to a parabolic equation for A , e.g., theinitial condition of A is used as Z in [DK90, Sec. 6.3]). The mollification operator χ ε : Ω k → Ω k also depends on our global section (or equivalently, on Z ).If we endow Ω k with the distance coming from its natural L Hilbert spacestructure then, for any g ∈ G ∞ = C ∞ ( T , G ) , the adjoint action Ad g : Ω k → Ω k isan isometry with the covariance properties Ad g ( A − Z ) = A g − Z g and Ad g d A ω = d A g Ad g ω . Finally, recall that F A is a two-form in Ω , and satisfies Ad g F A = F A g .With these preliminaries in mind, for any ξ ε ∈ C ∞ ([ , T ] , Ω ) , we rewrite thePDE (2.1) as ∂ t A = − d ∗ A F A − d A d ∗ A ( A − Z ) + ξ ε + C ( A − Z ) , A ( ) = a ∈ A , ain results where C ∈ R is a constant. Note that the right and left-hand sides take values in Ω . For a time-dependent gauge transformation g ∈ C ∞ ([ , T ] , G ∞ ) , we have that B def = A g satisfies ∂ t B = Ad g ∂ t A − d B [( ∂ t g ) g − ] . In particular, if g satisfies ( ∂ t g ) g − = d ∗ B ( Z g − Z ) , (2.2)then B solves ∂ t B = − d ∗ B F B − d B d ∗ B ( B − Z ) + Ad g ξ ε + C ( B − Z g ) , B ( ) = a g ( ) ∈ A . (2.3)The claimed gauge covariance of (2.1) is then a consequence of the non-trivial factthat one can choose the constant C in such a way that, as ε → , B converges tothe same limit in law as the SPDE (2.1) started from a g ( ) , i.e. ∂ t ˜ A = − d ∗ ˜ A F ˜ A − d ˜ A d ∗ ˜ A ( ˜ A − Z ) + ξ ε + C ( ˜ A − Z ) , ˜ A ( ) = a g ( ) ∈ A . (2.4)We now make this statement precise. Written in coordinates, the equations for thegauge transformed system are given by ∂ t B i = ∆ B i + gξ εi g − + CB i + C ( ∂ i g ) g − (2.5) + [ B j , ∂ j B i − ∂ i B j + [ B j , B i ]] , B ( ) = a g ( ) ∈ Ω α , ( ∂ t g ) g − = ∂ j (( ∂ j g ) g − ) + [ B j , ( ∂ j g ) g − ] , g ( ) ∈ G ,α . The desired gauge covariance is then stated as follows.
Theorem 2.8 (i) For every space-time mollifier χ there exists a unique ε -independent ¯ C ∈ L G ( g , g ) with the following property. For every C ∈ L G ( g , g ) , a ∈ Ω α , and g ( ) ∈ G ,α , if ( B, g ) is the solution to (2.5) and ( ¯ A, ¯ g ) is the solution to ∂ t ¯ A i = ∆ ¯ A i + χ ε ∗ ( ¯ gξ i ¯ g − ) + C ¯ A i + ( C − ¯ C )( ∂ i ¯ g ) ¯ g − (2.6) + [ ¯ A j , ∂ j ¯ A i − ∂ i ¯ A j + [ ¯ A j , ¯ A i ]] , ¯ A ( ) = a g ( ) , ( ∂ t ¯ g ) ¯ g − = ∂ j (( ∂ j ¯ g ) ¯ g − ) + [ ¯ A j , ( ∂ j ¯ g ) ¯ g − ] , ¯ g ( ) = g ( ) ,then ¯ A and B converge in probability to the same limit in Ω sol as ε → .(ii) If χ is a non-anticipative mollifier and ¯ C is as stated in item (i), then thesolution A to (2.1) with C = ¯ C is independent of χ . As discussed above, B = A g (i.e. B is pathwise gauge equivalent to A ) for anychoice of C . On the other hand, if χ is non-anticipative, then χ ε ∗ ( ¯ gξ i ¯ g − ) is equalin law to ξ εi by Itô isometry since ¯ g is adapted, so that when C = ¯ C , the law of ¯ A does not depend on ¯ g anymore and ¯ A is equal in law to the process ˜ A definedin (2.4), obtained by starting the dynamics for A from a g ( ) . The theorem thereforeproves the desired form of gauge covariance for the choice C = ¯ C . ain results Again, as in Remark 2.7 it suffices to prove Theorem 2.8 in the case ofa simple Lie algebra for which one has ¯ C ∈ R . In this case, for non-anticipative χ , ¯ C = − λ lim ε ↓ R dz χ ε ( z )( K ∗ K ε )( z ) , where K is the heat kernel, K ε = χ ε ∗ K ,and λ < is such that λ id g is the quadratic Casimir in the adjoint representation.We finally turn to the associated Markov process on gauge orbits. To state the wayin which our Markov process is canonical we introduce a particular class of Ω α -valued processes which essentially captures the “nice” ways to run the SPDE (2.1)and restart it from different representatives of gauge orbits. For an interval I ⊂ R and a metric space X , let D ( I, X ) denote the Skorokhod space of càdlàg functions A : I → X . For the remainder of this section, by a “white noise” we mean a pairof i.i.d. g -valued white noises ξ = ( ξ , ξ ) on R × T . Definition 2.10
Setting ˆΩ α def = Ω α ∪ { } , a probability measure µ on D ( R + , ˆΩ α ) is called generative if there exists a filtered probability space ( O , F , ( F t ) t ≥ , P ) supporting a white noise ξ for which the filtration ( F t ) t ≥ is admissible (i.e. ξ isadapted to ( F t ) t ≥ and ξ ↾ [ t, ∞ ) is independent of F t for all t ≥ ), and a randomvariable A : O → D ( R + , ˆΩ α ) with the following properties.1. The law of A is µ and A ( ) is F -measurable.2. For any ≤ s ≤ t , let Φ s,t : ˆΩ α → ˆΩ α denote the (random) solution mapin the ε → limit of (2.1) with a non-anticipative mollifier χ and constant C = ¯ C from part (i) of Theorem 2.8. There exists a sequence of stoppingtimes ( σ j ) ∞ j =0 , such that σ = 0 almost surely and, for all j ≥ ,(a) if σ j = ∞ , then σ j +1 = ∞ ,(b) if σ j < ∞ and A ( σ j ) = , then σ j +1 = σ j ; if A ( σ j ) = then σ j < σ j +1 and A ( t ) = Φ σ j ,t ( A ( σ j )) for all t ∈ [ σ j , σ j +1 ) ,(c) if σ j +1 < ∞ , then there exists an F σ j +1 -measurable random variable g j : O → G ,α such that A ( σ j +1 ) g j = Φ σ j ,σ j +1 ( A ( σ j )) . (We use theconvention g = for all g ∈ G ,α .)3. Let T ∗ def = inf { t ≥ | A ( t ) = } . If T ∗ < ∞ , then A ≡ on [ T ∗ , ∞ ) andfor any non-decreasing sequence of stopping times τ n ր T ∗ lim n →∞ inf g ∈ G ,α | A ( τ n ) g | α = ∞ . If there exists a ∈ ˆΩ α such that A ( ) = a almost surely, then we call a the initialcondition of µ . Remark 2.11
In the setting of Definition 2.10, if B : O → ˆΩ α is F s -measurable,then t Φ s,t ( B ) is adapted to ( F t ) t ≥ . In particular, the conditions on the process A imply that A is adapted to ( F t ) t ≥ .Denote ˆ O α def = O α ∪ { } and let π : ˆΩ α → ˆ O α denote the projection map.Note that if µ is generative, then the pushforward π ∗ µ is a probability measure Note that Φ exists and is independent of the choice of χ by part (ii) of Theorem 2.8. onstruction of the state space on C ( R + , ˆ O α ) (rather than just on D ( R + , ˆ O α ) ) thanks to items 2b, 2c, and 3 ofDefinition 2.10. With these notations, the Markov process on the space of gaugeorbits announced in the introduction is given by the following. Theorem 2.12 (i) For every a ∈ ˆΩ α , there exists a generative probability mea-sure µ with initial condition a . Moreover, one can take in Definition 2.10 ( F t ) t ≥ to be the filtration generated by any white noise and the process A itself to be Markov.(ii) There exists a unique ˆ O α -valued Markov process X such that, for every x ∈ ˆ O α , a ∈ x , and generative probability measure µ with initial condition a , the pushforward π ∗ µ is the law of X x . The aim of this section is to find a space of distributional -forms and a corre-sponding group of gauge transformations which can be used to construct the statespace for our Markov process. We would like our space to be sufficiently large tocontain -forms with components that “look like” a free field, but sufficiently smallthat there is a meaningful notion of integration along smooth enough curves. Ourspace of -forms is a strengthened version of that constructed in [Che19], the maindifference being that we do not restrict our notion of integration to axis-parallelpaths. Let X denote the set of oriented line segments in T of length at most . Specifically,denoting B r def = { v ∈ R | | v | ≤ r } , we define X def = T × B / (first coordinateis the initial point, second coordinate is the direction). We fix for the remainder ofthis section a Banach space E . Definition 3.1
We say that ℓ = ( x, v ) , ¯ ℓ = ( ¯ x, ¯ v ) ∈ X are joinable if ¯ x = x + v and there exist w ∈ R and c, ¯ c ∈ [ − , ] such that | w | = 1 , v = cw , ¯ v = ¯ cw , and | c + ¯ c | ≤ . In this case, we denote ℓ ⊔ ¯ ℓ def = ( x, ( c + ¯ c ) w ) ∈ X . We say that afunction A : X → E is additive if A ( ℓ ⊔ ¯ ℓ ) = A ( ℓ ) + A ( ¯ ℓ ) for all joinable ℓ, ¯ ℓ ∈ X .Let Ω = Ω ( T , E ) denote the space of all measurable E -valued additive functionson X .Note that additivity implies that A ( x, ) = 0 for all x ∈ T and A ∈ Ω . Remark 3.2
For A ∈ Ω , one should think of A ( ℓ ) as the line integral along ℓ ofa homogeneous function on the tangent bundle of T . To wit, any measurablefunction B : T × R → E which is bounded on T × B and homogeneous inthe sense that B ( x, cv ) = cB ( x, v ) for all ( x, v ) ∈ T × R and c ∈ R , defines anelement A ∈ Ω by A ( x, v ) def = Z B ( x + tv, v ) d t . onstruction of the state space We will primarily be interested in the case that B is a -form, i.e., B ( x, v ) is linearin v , and we discuss this situation in Section 3.3. However, many definitions andestimates turn out to be more natural in the general setting of Ω .For ℓ = ( x, v ) ∈ X , let us denote by ℓ i def = x and ℓ f def = x + v the initial and finalpoint of ℓ respectively. We define a metric d on X by d ( ℓ, ¯ ℓ ) def = | ℓ i − ¯ ℓ i | ∨ | ℓ f − ¯ ℓ f | . For ℓ = ( x, v ) ∈ X , let | ℓ | def = | v | denote its length. Definition 3.3
We say that ℓ, ¯ ℓ ∈ X are far if d ( ℓ, ¯ ℓ ) > ( | ℓ | ∧ | ¯ ℓ | ) . Define thefunction ̺ : X → [ , ∞ ) by ̺ ( ℓ, ¯ ℓ ) def = ( | ℓ | + | ¯ ℓ | if ℓ, ¯ ℓ are far, | ℓ i − ¯ ℓ i | + | ℓ f − ¯ ℓ f | + Area ( ℓ, ¯ ℓ ) / otherwise,where Area ( ℓ, ¯ ℓ ) is the area of the convex hull of of the points ( ℓ i , ℓ f , ¯ ℓ f , ¯ ℓ i ) (whichis well-defined whenever ℓ, ¯ ℓ are not far). Remark 3.4 If ℓ, ¯ ℓ ∈ X are not far, then their lengths are of the same order, andArea ( ℓ, ¯ ℓ ) is of the same order as | ℓ | [ d ( ¯ ℓ i , ℓ ) + d ( ¯ ℓ f , ℓ ) ] , where, denoting ℓ = ( x, v ) ,we have set d ( y, ℓ ) def = inf t ∈ [ − , ] | x + tv − y | (note the set [ − , ] in the infimuminstead of [ , ] ). In particular, it readily follows that although ̺ isn’t a metric ingeneral, it is a semimetric admitting a constant C ≥ such that for all a, b, c ∈ X ̺ ( a, b ) ≤ C ( ̺ ( b, c ) + ̺ ( b, c )) . (3.1)For α ∈ [ , ] , we define the (extended) norm on Ω | A | α def = sup ̺ ( ℓ, ¯ ℓ ) > | A ( ℓ ) − A ( ¯ ℓ ) | ̺ ( ℓ, ¯ ℓ ) α . (3.2)We also write Ω α for the Banach space { A ∈ Ω | | A | α < ∞} equipped with thenorm | · | α . Remark 3.5
By additivity, any element of Ω extends uniquely to an additive func-tion on all line segments, not just those of length less than / and we will use thisextension in the sequel without further mention. However, the supremum in (3.2)is restricted to these “short” line segments. Remark 3.6
Since we know that A ∈ Ω vanishes on line segments of zero length,it follows that | A ( ℓ ) | ≤ | A | α | ℓ | α , so that despite superficial appearances (3.2) is anorm on Ω α and not just a seminorm.We now introduce several other (semi)norms which will be used in the sequel. onstruction of the state space Define the (extended) norm on Ω | A | α -gr def = sup | ℓ | > | A ( ℓ ) || ℓ | α . Let Ω α -gr denote the Banach space { A ∈ Ω | | A | α -gr < ∞} equipped with the norm | · | α -gr . Definition 3.8
We say that a pair ℓ, ¯ ℓ ∈ X form a vee if they are not far, have thesame length | ℓ | = | ¯ ℓ | , and have the same initial point ℓ i = ¯ ℓ i . Define the (extended)seminorm on Ω | A | α -vee def = sup ℓ =¯ ℓ | A ( ℓ ) − A ( ¯ ℓ ) | Area ( ℓ, ¯ ℓ ) α/ ,where the supremum is taken over all distinct ℓ, ¯ ℓ ∈ X forming a vee. Definition 3.9
For a line segment ℓ = ( x, v ) ∈ X , let us denote the associatedsubset of T by ι ( ℓ ) def = ι ( x, v ) def = { x + cv : c ∈ [ , ) } . For an integer n ≥ , an n -gon is a tuple P = ( ℓ , . . . , ℓ n ) ∈ X n such that • ℓ i = ℓ nf , and ℓ ji = ℓ j − f for all j = 2 , . . . , n , • ι ( ℓ j ) ∩ ι ( ℓ k ) = for all distinct j, k ∈ { , . . . , n } , and • ι ( ℓ ) ∪ . . . ∪ ι ( ℓ n ) has diameter at most .A -gon is called a triangle .Note that an n -gon P splits T into two connected components, one of whichis simply connected and we denote by ˚ P . We further note that this split allowsus to define when two n -gons have the same orientation. For measurable subsets X, Y ∈ T , let X △ Y denote their symmetric difference, and | X | denote theLebesgue measure of X . For n -gons P , P , let us denote | P | def = | ˚ P | and | P ; P | def = ( | ˚ P △ ˚ P | if P , P have the same orientation | ˚ P | + | ˚ P | otherwise ,which we observe defines a metric on the set of n -gons. Definition 3.10
Let P = ( ℓ , . . . , ℓ n ) be an n -gon. For A ∈ Ω , we denote A ( ∂P ) def = n X j =1 A ( ℓ j ) . For α ∈ [ , ] we define the quantities | A | α -tr def = sup | P | > | A ( ∂P ) || P | α/ , (3.3) onstruction of the state space where the supremum is taken over all triangles P with | P | > , and | A | α -sym def = sup | P ; ¯ P | > | A ( ∂P ) − A ( ∂ ¯ P ) || P ; ¯ P | α/ ,where the supremum is taken over all triangles P, ¯ P with | P ; ¯ P | > .The motivation behind each norm is the following. • The norm | · | α -vee facilitates the analysis of gauge transformations (Sec-tion 3.4). • The norm | · | α -sym is helpful in extending the domain of definition of A ∈ Ω α to a wider class of curves (Section 3.2). • The norm |·| α -tr is simpler but equivalent to |·| α -sym . Furthermore, the values A ( ∂P ) can be evaluated using Stokes’ theorem (e.g., as in Lemma 4.8).We show now that each of these norms, when combined with | · | α -gr , is equivalentto | · | α . Theorem 3.11
There exists C ≥ such that for all α ∈ [ , ] and A ∈ Ω C − | A | α ≤ | A | α -gr + | A | • ≤ C | A | α ,where • is any one of α -vee , α -tr , or α -sym . For the proof, we require the following lemmas.
Lemma 3.12
For α ∈ [ , ] and n ≥ , it holds that sup | P | > | A ( ∂P ) || P | α/ ≤ C n | A | α -tr ,where the supremum is taken over all n -gons P with | P | > , and where C def = 1 and for n ≥ C n def = C n − + ( C − / ( − α ) n − ) α/ ( C − / ( − α ) n − ) α/ . Proof.
This readily follows by induction, using the two ears theorem and the factthat C n is the optimal constant such that x α/ + C n − y α/ ≤ C n ( x + y ) α/ for all x, y ≥ . Lemma 3.13
There exists C ≥ such that for all α ∈ [ , ] and A ∈ Ω | A | α -tr ≤ | A | α -sym ≤ C | A | α -tr . onstruction of the state space Proof.
The first inequality is obvious by taking ¯ P in the definition of | A | α -sym asany degenerate triangle. For the second, let P , P be two triangles. We need onlyconsider the case that P , P are oriented in the same direction. Observe that thereexist k ≤ and Q , . . . , Q k , where each Q i is an n -gon with n ≤ , such that | ˚ P △ ˚ P | = P ki =1 | ˚ Q i | , and such that A ( ∂P ) − A ( ∂P ) = P ki =1 A ( ∂Q i ) . It thenfollows from Lemma 3.12 that | A ( ∂P ) − A ( ∂P ) | . | A | α -tr k X i =1 | ˚ Q i | α/ . | A | α -tr | ˚ P △ ˚ P | α/ ,as required. Proof of Theorem 3.11.
We show first C − | A | α ≤ | A | α -gr + | A | α -vee ≤ C | A | α . (3.4)The second inequality in (3.4) is clear (without even assuming that A is additive)since | ℓ | = ̺ ( ℓ, ¯ ℓ ) whenever | ¯ ℓ | = 0 , and for any ℓ, ¯ ℓ ∈ X forming a vee, we have ̺ ( ℓ, ¯ ℓ ) . Area ( ℓ, ¯ ℓ ) / .It remains to show the first inequality in (3.4). If ℓ, ¯ ℓ are far, then clearly | A ( ℓ ) − A ( ¯ ℓ ) | . ̺ ( ℓ, ¯ ℓ ) α | A | α -gr . Supposing now that ℓ, ¯ ℓ are not far, we want toshow that | A ( ℓ ) − A ( ¯ ℓ ) | . ̺ ( ℓ, ¯ ℓ ) α ( | A | α -gr + | A | α -vee ) . (3.5)Consider the line segment a with initial point ℓ i and endpoint ¯ ℓ f , and the linesegment ¯ a ∈ X such that ¯ a = ( ℓ i , c ( ¯ ℓ f − ℓ i )) for some c > and | ¯ a | = | ℓ | . Notethat it is possible that | a | > , and thus a / ∈ X , however A ( a ) still makes sense byadditivity of A . Observe that | a f − ¯ a f | . | ℓ f − ¯ ℓ f | and Area ( ℓ, ¯ a ) . Area ( ℓ, ¯ ℓ ) (for the latter, note that a is contained inside the convex hull of ℓ, ¯ ℓ , and that ¯ a is atmost twice the length of a ).Suppose first that ¯ a and ℓ form a vee. Then breaking up A ( a ) into A ( ¯ a ) and aremainder, we see by additivity of A that | A ( ℓ ) − A ( a ) | . | A | α -vee Area ( ℓ, ¯ ℓ ) α/ + | A | α -gr | ℓ f − ¯ ℓ f | α . Suppose now that ¯ a and ℓ do not form a vee. Then we have | ℓ | . | Area ( ¯ a, ℓ ) | , andthus | A ( ℓ ) − A ( a ) | . | A | α -gr ( | ℓ | α + | ℓ f − ¯ ℓ f | α ) . | A | α -gr ( Area ( ℓ, ¯ ℓ ) α/ + | ℓ f − ¯ ℓ f | α ) . By symmetry, one obtains | A ( ¯ ℓ ) − A ( a ) | . ( | A | α -vee + | A | α -gr )( Area ( ℓ, ¯ ℓ ) α/ + | ℓ i − ¯ ℓ i | α ) ,which proves (3.5).For the remaining inequalities, one can readily see that | A | α -vee . | A | α -gr + | A | α -tr , onstruction of the state space so that the claim follows if we can show that | A | α -tr . | A | α . (3.6)For this, consider a triangle P = ( ℓ , ℓ , ℓ ) and assume without loss of generalitythat | ℓ | ≥ | ℓ | ≥ | ℓ | . Suppose first that P is right-angled. If ℓ , ℓ are far, then P j =1 | ℓ j | . | P | / , while if ℓ , ℓ are not far, then ̺ ( ℓ , ℓ ) . | P | / . In eithercase, | A ( ∂P ) | ≤ | A | α | P | / . For general P , we can split P into two right-angledtriangles P , P with | P | + | P | = | P | and A ( ∂P ) = A ( ∂P ) + A ( ∂P ) and applythe previous case, which proves (3.6). The conclusion follows from Lemma 3.13.For A ∈ Ω and ℓ = ( x, v ) ∈ X , define the function ℓ A : [ , ] → E by ℓ A ( t ) def = A ( x, tv ) . Lemma 3.14
There exists a constant
C > such that for all α ∈ [ , ] , A ∈ Ω ,and ℓ, ¯ ℓ ∈ X forming a vee, one has | ℓ A | α -Höl ≤ | ℓ | α | A | α -gr , | ℓ A − ¯ ℓ A | α -Höl ≤ C Area ( ℓ, ¯ ℓ ) α/ | A | α . (3.7) Proof.
The first inequality is obvious by additivity of A . For the second, let ≤ s < t ≤ and denote by ℓ s,t the sub-segment of ℓ = ( x, v ) with initial point x + sv and final point x + tv . We claim that ̺ ( ℓ s,t , ¯ ℓ s,t ) . | t − s | / Area ( ℓ, ¯ ℓ ) / . (3.8)Indeed, observe that Area ( ℓ, ¯ ℓ ) ≍ | ℓ || ℓ f − ¯ ℓ f | as a consequence of the fact that | ℓ | = | ¯ ℓ | and ℓ i = ¯ ℓ i by the definition of “forming a vee”. One furthermore has theidentities | ℓ s,t | = | ¯ ℓ s,t | = | t − s || ℓ | , and | ℓ s,tf − ¯ ℓ s,tf | = t | ℓ f − ¯ ℓ f | . Hence, if ℓ s,t , ¯ ℓ s,t are far, then we must have | t − s || ℓ | . t | ℓ f − ¯ ℓ f | ≤ | ℓ f − ¯ ℓ f | ,and thus ̺ ( ℓ s,t , ¯ ℓ s,t ) = 2 | t − s || ℓ | . | t − s | / Area ( ℓ, ¯ ℓ ) / . On the other hand, if ℓ s,t , ¯ ℓ s,t are not far, then we must have t | ℓ f − ¯ ℓ f | . | t − s || ℓ | ,and thus t / | ℓ f − ¯ ℓ f | . | t − s | / | ℓ | / | ℓ f − ¯ ℓ f | / ≍ | t − s | / Area ( ℓ, ¯ ℓ ) / . It follows that ̺ ( ℓ s,t , ¯ ℓ s,t ) ≤ t | ℓ f − ¯ ℓ f | + | t − s | / Area ( ℓ, ¯ ℓ ) / . | t − s | / Area ( ℓ, ¯ ℓ ) / ,which proves (3.8). It follows that | ℓ A ( t ) − ℓ A ( s ) − ¯ ℓ A ( t ) + ¯ ℓ A ( s ) | = | A ( ℓ s,t ) − A ( ¯ ℓ s,t ) | . | t − s | α/ Area ( ℓ, ¯ ℓ ) α/ | A | α ,concluding the proof. onstruction of the state space In this subsection we show that any element A ∈ Ω α extends to a well-definedfunctional on sufficiently regular curves γ : [ , ] → T . Given that A ( γ ) shouldbe invariant under reparametrisation of γ , we first provide a way to measure theregularity of γ in a parametrisation invariant way, and later provide relations tomore familiar spaces of paths (namely paths in C ,β ).For a function γ : [ s, t ] → T , we denote by diam ( γ ) def = sup u,v ∈ [ s,t ] | γ ( u ) − γ ( v ) | the diameter of γ . We assume throughout this subsection that all functions γ : [ s, t ] → T under consideration have diameter at most .We call a partition of an interval [ s, t ] a finite collection of subintervals D = { [ t i , t i +1 ] | i ∈ { , . . . , n − }} , with t = s < t < . . . < t n − < t n = t , andwe write D ([ s, t ]) for the set of all partitions. For a function γ : [ s, t ] → T and D ∈ D ([ s, t ]) , let γ D be the piecewise affine interpolation of γ along D . Note that if γ is piecewise affine, then there exists D ∈ D ([ s, t ]) and elements ℓ i = ( x i , v i ) ∈ X such that, for u ∈ [ t i , t i +1 ] , one has γ ( u ) = x i + v i ( u − t i ) / ( t i +1 − t i ) . A ( γ ) isthen canonically defined by A ( γ ) = P i A ( ℓ i ) . (This is independent of the choiceof t i and ℓ i parametrising γ .) Definition 3.15
Let A ∈ Ω and γ : [ , ] → T d . We say that A extends to γ if thelimit γ A ( t ) def = lim | D |→ A ( γ D ↾ [ ,t ] ) (3.9)exists for all t ∈ [ , ] , where D ∈ D ([ , ]) and | D | def = max [ a,b ] ∈ D | b − a | .The following definition provides a convenient, parametrisation invariant wayto determine if a given A ∈ Ω extends to γ . Definition 3.16
Let γ : [ , ] → T be a function. The triangle process associatedto γ is defined to be the function P defined on [ , ] , taking values in the set oftriangles, such that P sut is the triangle formed by ( γ ( s ) , γ ( u ) , γ ( t )) .For two functions γ, ¯ γ : [ , ] → T , and a subinterval [ s, t ] ⊂ [ , ] , define | γ ; ¯ γ | [ s,t ] def = sup u ∈ [ s,t ] | P sut ; ¯ P sut | / ,where P, ¯ P are the triangle processes associated to γ, ¯ γ respectively. For α ∈ [ , ] ,define further | γ ; ¯ γ | α ; [ s,t ] def = sup D ∈D ([ s,t ]) X [ a,b ] ∈ D | γ ; ¯ γ | α [ a,b ] . We denote | γ | α ; [ s,t ] def = | γ ; ¯ γ | α ; [ s,t ] where ¯ γ is any constant path. We drop thereference to the interval [ s, t ] whenever [ s, t ] = [ , ] .We note the following basic properties of |· ; ·| α : onstruction of the state space • |· ; ·| α is symmetric and satisfies the triangle inequality but defines only apseudometric rather than a metric since any two affine paths are at distance from each other. • The map α
7→ | γ ; ¯ γ | α is decreasing in α for any γ, ¯ γ : [ , ] → T . • For a typical smooth curve, | P sut | is of order | t − s | (cf. (3.13) below). Itfollows that | γ | α < ∞ for all smooth γ : [ , ] → T if and only if α ≥ .Recall (see, e.g., [FV10, Def.1.6]) that a control is a continuous, super-additivefunction ω : { ( s, t ) | ≤ s ≤ t ≤ } → R + such that ω ( t, t ) = 0 . Here super-additivity means that ω ( s, t ) + ω ( t, u ) ≤ ω ( s, u ) for any s ≤ t ≤ u . Lemma 3.17
Let γ, ¯ γ ∈ C ([ , ] , T ) such that | γ ; ¯ γ | α < ∞ . Then ω : ( s, t ) γ ; ¯ γ | α ; [ s,t ] is a control. The proof of Lemma 3.17 follows in the same way as the more classical statementthat ( s, t )
7→ | γ | pp -var ; [ s,t ] is a control, see e.g. the proof of [FV10, Prop. 5.8] (notethat continuity is the only subtle part). Theorem 3.18
Let ≤ α < ¯ α ≤ and denote θ def = ¯ α/α . Let A ∈ Ω with | A | ¯ α -sym < ∞ and γ ∈ C ([ , ] , T ) such that | γ | α < ∞ . Then A extends to γ andfor any partition D of [ , ] | A ( γ D ) − A ( γ ) | ≤ θ ζ ( θ ) | A | ¯ α -sym X [ s,t ] ∈ D | γ | θα ; [ s,t ] , (3.10) where ζ is the classical Riemann zeta function. Let ¯ γ ∈ C ([ , ] , T ) be anotherpath such that | ¯ γ | α < ∞ . Then | A ( γ ) − A ( ¯ γ ) | ≤ | A ( ℓ ) − A ( ¯ ℓ ) | + 2 θ ζ ( θ ) | A | ¯ α -sym | γ ; ¯ γ | θα , (3.11) where ℓ, ¯ ℓ ∈ X are the line segments connecting γ ( ) , γ ( ) and ¯ γ ( ) , ¯ γ ( ) respec-tively.Proof. Define ω ( s, t ) def = | γ ; ¯ γ | α ; [ s,t ] , which we note is a control by Lemma 3.17.Let D be a partition of [ , ] . We will apply Young’s partition coarsening argumentto show that | A ( γ D ) − A ( ¯ γ D ) | ≤ | A ( ℓ ) − A ( ¯ ℓ ) | + | γ ; ¯ γ | θα θ ζ ( θ ) | A | ¯ α -sym . (3.12)Let n denote the number of points in D . If n = 2 , then the claim is obvious. If n ≥ ,then by superadditivity of ω there exist two adjacent subintervals [ s, u ] , [ u, t ] ∈ D such that ω ( s, t ) ≤ ω ( , ) / ( n − ) . Let P, ¯ P denote the triangle process associatedwith γ, ¯ γ respectively. Observe that | A ( ∂P sut ) − A ( ∂ ¯ P sut ) | ≤ | γ ; ¯ γ | ¯ α [ s,t ] | A | ¯ α -sym ≤ ω ( s, t ) θ | A | ¯ α -sym ≤ ( ω ( , ) / ( n − )) θ | A | ¯ α -sym . onstruction of the state space Merging the intervals [ s, u ] , [ u, t ] ∈ D into [ s, t ] yields a coarser partition D ′ andwe see that | A ( γ D ) − A ( ¯ γ D ) − ( A ( γ D ′ ) − A ( ¯ γ D ′ ) ) | = | A ( ∂P sut ) − A ( ∂ ¯ P sut ) |≤ ( ω ( , ) / ( n − )) θ | A | ¯ α -sym . Proceeding inductively, we obtain (3.12). It remains only to show that (3.9) existsfor t = 1 and satisfies (3.10). By Lemma 3.17, we have lim ε → sup | D | <ε X [ s,t ] ∈ D | γ | θα ; [ s,t ] = 0 . Observe that if D ′ is a refinement of D , then we can apply the uniform bound (3.12)to every [ s, t ] ∈ D to obtain | A ( γ D ) − A ( γ D ′ ) | ≤ θ ζ ( θ ) | A | ¯ α -sym X [ s,t ] ∈ D | γ | θα ; [ s,t ] ,from which the existence of (3.9) and the bound (3.10) follow.For a metric space ( X, d ) and p ∈ [ , ∞ ) , recall that the p -variation | x | p -var ofa path x : [ , ] → X is given by | x | pp -var def = sup D ∈D ([ , ]) X [ s,t ] ∈ D d ( x ( s ) , x ( t )) p . Our interest in p -variation stems from the Young integral [You36, FH14] whichensures that ODEs driven by finite p -variation paths are well-defined. Corollary 3.19
Let ≤ α < ¯ α ≤ , η ∈ ( , ] , and p ≥ η . Consider γ ∈C ([ , ] , T ) with | γ | α < ∞ and A ∈ Ω with | A | ¯ α -sym + | A | η -gr < ∞ . Then | γ A | p -var ≤ | A | η -gr | γ | ηpη -var + 2 ¯ α/α ζ ( ¯ α/α ) | A | ¯ α -sym | γ | ¯ α/αα . Proof.
For any [ s, t ] ⊂ [ , ] , (3.11) implies that | γ A ( t ) − γ A ( s ) | ≤ | A | η -gr | γ ( t ) − γ ( s ) | η + 2 ¯ α/α ζ ( ¯ α/α ) | A | ¯ α -sym | γ | ¯ α/αα ; [ s,t ] ,from which the conclusion follows by Minkowski’s inequality.The following result provides a convenient (now parametrisation dependent)way to control the quantity | γ ; ¯ γ | α . For β ∈ [ , ] , let C ,β ([ , ] , T ) denote thespace of differentiable functions γ : [ , ] → T with ˙ γ ∈ C β . Recall that | · | ∞ denotes the supremum norm. Proposition 3.20
There exists
C > such that for all α ∈ [ , ] , β ∈ [ α − , ] ,and κ ∈ [ − αα ( β ) , ] , and γ, ¯ γ ∈ C ,β ([ , ] , T ) , it holds that | γ ; ¯ γ | α ≤ C h ( | ˙ γ | ∞ + | ˙¯ γ | ∞ )( | ˙ γ | β -Höl + | ˙¯ γ | β -Höl ) κ | γ − ¯ γ | − κ ∞ i α/ . onstruction of the state space Proof.
Let P, ¯ P denote the triangle process associated to γ, ¯ γ respectively. For ≤ s < u < t ≤ , observe that | P sut | ≤ | γ ( t ) − γ ( s ) | (cid:12)(cid:12)(cid:12) γ ( u ) − γ ( s ) − | u − s || t − s | ( γ ( t ) − γ ( s )) (cid:12)(cid:12)(cid:12) ≤ | t − s || ˙ γ | ∞ (cid:12)(cid:12)(cid:12) | t − s | Z us Z ts | ˙ γ ( r ) − ˙ γ ( q ) | d q d r (cid:12)(cid:12)(cid:12) ≤ | t − s | β | ˙ γ | ∞ | ˙ γ | β -Höl . (3.13)Furthermore, | P sut ; ¯ P sut | . X q = r ( | γ ( q ) − γ ( r ) | + | ¯ γ ( q ) − ¯ γ ( r ) | )( | γ ( q ) − ¯ γ ( q ) | + | γ ( r ) − ¯ γ ( r ) | ) ,where the sum is over all -subsets { q, r } of { s, u, t } , whence | P sut ; ¯ P sut | . | t − s | ( | ˙ γ | ∞ + | ˙¯ γ | ∞ ) | γ − ¯ γ | ∞ . (3.14)Interpolating between (3.13) and (3.14), we have for any κ ∈ [ , ] | P sut ; ¯ P sut | . ( | ˙ γ | ∞ + | ˙¯ γ | ∞ )( | ˙ γ | β -Höl + | ˙¯ γ | β -Höl ) κ | t − s | κ + βκ | γ − ¯ γ | − κ ∞ . The conclusion following taking κ ≥ − αα ( β ) so that α ( κ + βκ ) / ≥ .We end this subsection with a result on the continuity in p -variation of γ A jointlyin ( A, γ ) ∈ Ω α × C ,β ([ , ] , T ) . For β ∈ [ , ] , a ball in C ,β is any set of theform { γ ∈ C ,β ([ , ] , T ) | | ˙ γ | ∞ + | ˙ γ | β -Höl ≤ R } for some R ≥ . Proposition 3.21
Let α ∈ ( , ] , p > α , and β ∈ ( α − , ] . There exists δ > such that for all A, ¯ A ∈ Ω α | γ A − ¯ γ ¯ A | p -var . | A − ¯ A | α + | A | α | γ − ¯ γ | δ ∞ uniformly over γ, ¯ γ in balls of C ,β .Proof. Note that | γ | -var is trivially bounded by | ˙ γ | ∞ . Furthermore, for ¯ α ∈ [ , ] ,it follows from Proposition 3.20 that | γ | ¯ α is uniformly bounded on balls in C , ¯ β with ¯ β = α − . As a consequence (using that β > α − ), it follows from Corollary 3.19that | γ A | α -var . | A | α uniformly on balls in C ,β .On the other hand, by (3.11) and Proposition 3.20 (using again that β > α − ),there exists ε > such that | γ A − ¯ γ A | ∞ . | A | α | γ − ¯ γ | ε ∞ uniformly over balls in C ,β . Applying the interpolation estimate for p > α and x : [ , ] → E | x | p -var ≤ ( | x | α -var ) αp ( | x | ∞ ) − αp , onstruction of the state space it follows that for some δ > | γ A − ¯ γ A | p -var . | A | α | γ − ¯ γ | δ ∞ uniformly on balls in C ,β .Finally, it follows again from Corollary 3.19 and Proposition 3.20 that | γ A − γ ¯ A | α -var = | γ A − ¯ A | α -var . | A − ¯ A | α uniformly over balls in C ,β , from which the conclusion follows. -forms Recall that Ω C def = Ω C ( T , E ) denotes the Banach space of continuous E -valued -forms. Following Remark 3.2, there exists a canonical map ı : Ω C → Ω -gr (3.15)defined by ıA ( x, v ) def = Z
10 2 X i =1 A i ( x + tv ) v i d t ,which is injective and satisfies | ıA | -gr ≤ | A | ∞ . Definition 3.22
For α ∈ [ , ] , let Ω α and Ω α -gr denote the closure of ı ( Ω C ∞ ) in Ω α and Ω α -gr respectively. Remark 3.23
Recalling notation from Section 1.5, it is easy to see that ı embeds Ω C α/ and Ω C ,α/ continuously into Ω α and Ω α respectively (and that the exponent α/ is sharp in the sense that Ω C β and Ω C ,β do not embed into Ω α and Ω α for any β < α/ ). Remark 3.24
Note that since any element of Ω C ∞ can be approximated by atrigonometric polynomial with rational coefficients, Ω α is a separable Banach spacewhenever E is separable.We now construct a continuous, linear map π : Ω α -gr → Ω C α − which is a leftinverse to ı and which we will use to classify the space Ω α -gr . Consider α ∈ ( , ] and A ∈ Ω α -gr . For ℓ = ( z, w ) ∈ X , v ∈ B / , and s ∈ [ , ] , define X s,ℓv ∈C α -Höl ([ , ] , E ) by X s,ℓv ( t ) def = A ( z + sw, tv ) (note that | X s,ℓv | α -Höl ≤ | v | α | A | α -gr by Lemma 3.14). In a similar way, for ψ ∈ C ( T ) , consider Y s,ℓv ∈ C ([ , ] , R ) given by Y s,ℓv ( t ) def = ψ ( z + sw + tv ) . We define the E -valued distribution π ℓ,v A ∈D ′ ( T , E ) by h π ℓ,v A, ψ i def = | v w − w v | Z d s Z Y s,ℓv ( t ) d X s,ℓv ( t ) , onstruction of the state space where the inner integral is in the Young sense.To motivative this definition, consider the parallelogram P ( ℓ, v ) def = { z + sw + tv | s, t ∈ [ , ] } ⊂ T . Note that the factor | v w − w v | is the area of P ( ℓ, v ) . By additivity of A , onehas the following basic properties:1. if ψ has support inside P ( ℓ, v ) ∩ P ( ¯ ℓ, v ) , then h π ℓ,v A, ψ i = h π ¯ ℓ,v A, ψ i ,2. if ℓ and ¯ ℓ are joinable (in which case P ( ℓ ⊔ ¯ ℓ, v ) = P ( ℓ, v ) ∪ P ( ¯ ℓ, v ) and P ( ℓ, v ) , P ( ¯ ℓ, v ) intersect only on their boundaries) then π ℓ ⊔ ¯ ℓ,v A = π ℓ,v A + π ¯ ℓ,v A ,3. similarly, if cv, ¯ cv, ( c + ¯ c ) v ∈ B / for some v ∈ R and c, ¯ c ∈ R , then π ℓ, ( c +¯ c ) v A = π ℓ,cv A + π ( z + cv,w ) , ¯ cv A .With these considerations, one should interpret h π ℓ,v A, ψ i as the integral of A against ψ inside P ( ℓ, v ) in the direction v . In fact, for A ∈ Ω C , one has the identity h π ℓ,v ıA, ψ i = X i =1 v i Z P ( ℓ,v ) A i ( x ) ψ ( x ) d x . (3.16)Finally, we define π : Ω α -gr → Ω C α − by setting ( πA ) i def = X n =1 π ℓ n , e i A ,where ℓ n ∈ X for n = 1 , . . . , are such that P ( ℓ n , e i ) partition T ≃ [ − , ) into squares (note that this definition does not depend on the particular choice).One can show that | πA | Ω C α − . | A | α -gr (see [Che19, Prop. 3.21]) and that π is aleft inverse of ı in the sense that, for all A ∈ Ω C , π ( ıA ) = A as distributions.Observe that for A ∈ ı Ω C , ℓ ∈ X , and v ∈ B / , (3.16) implies the linearityproperty h π ℓ,v A, ψ i = X i =1 v i h ( πA ) i , ψ i (3.17)for all ψ ∈ C ( T ) with support in P ( ℓ, v ) . The following result shows that thisproperty essentially characterises the space Ω α -gr . Furthermore, we see that π isinjective on Ω α -gr and one has a direct way to recover A ∈ Ω α -gr from πA throughmollifier approximations. In particular, we can identify Ω α -gr (and a fortiori Ω α )with a subspace of Ω C α − .Recall the definition of a mollifier from Section 1.5. For ε ∈ ( , ] , a mollifier χ on R , and A ∈ Ω α -gr , define A χ,ε ∈ Ω C ∞ by A χ,εi ( z ) def = h ( πA ) i , χ ε ( · − z ) i . We then have the following characterisation of these spaces. onstruction of the state space
Let α ∈ ( , ) and A ∈ Ω α -gr . Then the following are equivalent:(i) A ∈ Ω α -gr ,(ii) A is a continuous function on X , lim ε → sup | ℓ | <ε | ℓ | − α | A ( ℓ ) | = 0 , (3.18) and (3.17) holds for every ℓ ∈ X , v ∈ B / , and ψ ∈ C ( T ) with support in P ( ℓ, v ) .(iii) for every δ > there exists ε > such that | A − ıA χ,ε | α -gr ≤ δ for allmollifiers χ on R .Proof. The implication (iii) ⇒ (i) is obvious. We now show (i) ⇒ (ii). Let A ∈ Ω α -gr and δ > . Consider a sequence ( A n ) n ≥ in Ω C ∞ such that lim n →∞ | ıA n − A | α -gr = 0 . Since (3.17) holds for every ıA n , we see by continuity that (3.17) holdsfor A . Define functions B n : X → E by B n ( ℓ ) def = ( | ℓ | − α ıA n ( ℓ ) if | ℓ | > , otherwise.We define B : X → E in the same way with ıA n replaced by A . Observe that,since A n ∈ Ω C ∞ , B n is a continuous function on X . Furthermore, lim n →∞ | ıA n − A | α -gr = 0 is equivalent to lim n →∞ sup ℓ ∈X | B n ( ℓ ) − B ( ℓ ) | = 0 . Hence B is continuous on X , from which continuity of A and (3.18) follow. Thiscompletes the proof of (ii).It remains to show (ii) ⇒ (iii). Suppose that (ii) holds and let δ > . Define B : X → E as above, which is uniformly continuous by (ii) and compactness of X .In particular, there exists ε > such that | B ( y, v ) − B ( x, v ) | ≤ δ for all ( x, v ) ∈ X and y ∈ T such that | x − y | ≤ ε . Hence, for any ℓ = ( x, v ) ∈ X and mollifier χ on T , (cid:12)(cid:12)(cid:12) A ( ℓ ) − Z T χ ε ( h ) A ( x + h, v ) d h (cid:12)(cid:12)(cid:12) ≤ δ | ℓ | α . For ℓ = ( x, v ) ∈ X and a mollifier χ , define χ εℓ ∈ C ∞ ( T ) by χ εℓ ( z ) def = Z χ ε ( z − x − tv ) d t . Note that the support of χ εℓ shrinks to ι ( ℓ ) as ε → . Thus, taking | ℓ | < and ε sufficiently small, we can find ¯ ℓ ∈ X such that P ( ¯ ℓ, v ) contains the support of χ εℓ .Hence Z T χ ε ( h ) A ( x + h, v ) d h = h π ¯ ℓ,v A, χ εℓ i = X i =1 v i h ( πA ) i , χ εℓ i = ıA χ,ε ( ℓ ) , onstruction of the state space where we used (3.17) in the second equality, and the first and third equalities followreadily from definitions and additivity of A . We conclude that, for any ℓ ∈ X andmollifier χ , | A ( ℓ ) − ıA χ,ε ( ℓ ) | ≤ δ | ℓ | α , from which (iii) follows. For the remainder of the section, we fix a compact Lie group G with Lie algebra g .We equip g with an arbitrary norm and henceforth take E = g as our Banach space.Since G is compact, we can assume without loss of generality that G (resp. g ) is aLie subgroup of unitary matrices (resp. Lie subalgebra of anti-Hermitian matrices),so that both G and g are embedded in some normed linear space F of matrices. For g ∈ G , we denote by Ad g : g → g the adjoint action Ad g ( X ) = gXg − .For α ∈ [ , ] and a function g : T → F , recall the definition of the seminorm | g | α -Höl and norm | g | ∞ . We denote by G α the subset C α ( T , G ) , which we note isa topological group. Definition 3.26
Let α ∈ ( , ] , A ∈ Ω α -gr , β ∈ ( , ] with α + β > , and g ∈ G β .Define A g ∈ Ω by A g ( ℓ ) def = Z ( Ad g ( x + tv ) d ℓ A ( t ) − [ d g ( x + tv ) ] g − ( x + tv ) ) ,where ℓ = ( x, v ) ∈ X , and where both terms make sense as g -valued Youngintegrals since α + β > and β > . In the case α > , for A, ¯ A ∈ Ω α -gr we write A ∼ ¯ A if there exists g ∈ G α such that A g = ¯ A .Note that, in the case that A is a continuous -form and g is C , we have d ℓ A ( t ) = A ( x + tv )( v ) d t , hence A g ( x ) = Ad g ( x ) A ( x ) − [ d g ( x ) ] g − ( x ) ,as one expects from interpreting A as a connection. However, in the interpretationof A as a -form, the more natural map is A A g − g , which is linear and makessense for any β ∈ ( , ] such that α + β > (here is an element of Ω α -gr and,despite the notation, g is in general non-zero).The main result of this subsection is the following. Theorem 3.27
Let β ∈ ( , ] and α ∈ ( , ] such that α + β > and α + β > .Then the map ( A, g ) A g is a continuous map from Ω α × G β (resp. Ω α -gr × G β )into Ω α ∧ β (resp. Ω α ∧ β -gr ). If α ≤ β , then this map defines a left-group action, i.e., ( A h ) g = A gh . We give the proof of Theorem 3.27 at the end of this subsection. We begin byanalysing the case A = 0 . Proposition 3.28
Let α ∈ ( , ] and g ∈ G α . Then | g | α . | g | α -Höl ∨ | g | α -Höl ,where the proportionality constant depends only on α . onstruction of the state space For the proof of Proposition 3.28, we require several lemmas.
Lemma 3.29
Let α ∈ [ , ] , g ∈ G α , and ℓ = ( x, v ) , ¯ ℓ = ( ¯ x, ¯ v ) ∈ X forming avee. Consider the path ℓ g : [ , ] → G given by ℓ g ( t ) = g ( x + tv ) , and similarlyfor ¯ ℓ g . Then | ℓ g | α -Höl . | ℓ | α | g | α -Höl (3.19) and | ℓ g − ¯ ℓ g | α/ -Höl . | g | α -Höl Area ( ℓ, ¯ ℓ ) α/ (3.20) for universal proportionality constants.Proof. We have | ℓ g ( t ) − ℓ g ( s ) | ≤ | g | α -Höl | t − s | α | ℓ | α , which proves (3.19). For (3.20),we have | ℓ g ( t ) − ℓ g ( s ) − ¯ ℓ g ( t ) + ¯ ℓ g ( s ) | ≤ | g | α -Höl [ ( t α | ℓ f − ¯ ℓ f | α ) ∧ ( | t − s | α | ℓ | α ) ] . | g | α -Höl | t − s | α/ Area ( ℓ, ¯ ℓ ) α/ ,where in the second inequality we used interpolation and the fact that Area ( ℓ, ¯ ℓ ) ≍| ℓ f − ¯ ℓ f || ℓ | . Lemma 3.30
Let α ∈ ( , ] and g ∈ G α . Then | g | α -gr . | g | α -Höl ∨ | g | α -Höl , wherethe proportionality constant depends only on α .Proof. Let ℓ = ( x, v ) ∈ X . Then by (3.19) and Young’s estimate | g ( ℓ ) | = (cid:12)(cid:12)(cid:12) Z d g ( x + tv ) g − ( x + tv ) (cid:12)(cid:12)(cid:12) . ( | g | α -Höl | ℓ | α ) | ℓ | α | g | α -Höl ,which implies the claim. Lemma 3.31
Let α ∈ ( , ] and g ∈ G α . Then | g | α -vee . | g | α -Höl ∨ | g | α -Höl ,where the proportionality constant depends only on α .Proof. Let ℓ = ( x, v ) , ¯ ℓ = ( x, ¯ v ) ∈ X form a vee. Then, denoting Y t def = g − ( x + tv ) , ¯ Y t def = g − ( x + t ¯ v ) , and X t def = g ( x + tv ) , ¯ X t def = g ( x + t ¯ v ) , we have | g ( ℓ ) − g ( ¯ ℓ ) | = (cid:12)(cid:12)(cid:12) Z Y t d X t − Z ¯ Y t d ¯ X t (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) Z ( Y t − ¯ Y t ) d X t (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Z ¯ Y t d( X t − ¯ X t ) (cid:12)(cid:12)(cid:12) . Using (3.20), (3.19), Young’s estimate, and the fact that Y = ¯ Y , we have (cid:12)(cid:12)(cid:12) Z ( Y t − ¯ Y t ) d X t (cid:12)(cid:12)(cid:12) . | Y − ¯ Y | α/ -Höl | X | α -Höl . | g | α -Höl Area ( ℓ, ¯ ℓ ) α/ | ℓ | α onstruction of the state space and (cid:12)(cid:12)(cid:12) Z ¯ Y t d( X t − ¯ X t ) (cid:12)(cid:12)(cid:12) . ( | ¯ Y | α -Höl ) | X − ¯ X | α/ -Höl . ( | ℓ | α | g | α -Höl ) | g | α -Höl Area ( ℓ, ¯ ℓ ) α/ ,thus concluding the proof. Proof of Proposition 3.28.
Combining the equivalence of norms | · | α ≍ | · | α -gr + | · | α -vee from Theorem 3.11 with Lemmas 3.30 and 3.31 yields the proof.For the lemmas which follow, recall that the quantity A g − g makes sense forall A ∈ Ω α -gr and g ∈ G β provided that α, β ∈ ( , ] with α + β > . Lemma 3.32
Let α, β ∈ ( , ] such that α + β > , A ∈ Ω α -gr , and g ∈ G β . Then | A g − g − A | α -gr . ( | g − | ∞ + | g | β -Höl ) | A | α -gr , (3.21) where the proportionality constant depends only on α and β .Proof. Let ℓ = ( x, v ) ∈ X , A ∈ Ω , and g : T → G . Using notation fromLemma 3.29, note that ( A g − g − A )( ℓ ) = Z (Ad ℓ g ( t ) − ) d ℓ A ( t ) . (3.22)Using (3.7), (3.19), and Young’s estimate, we obtain | A g ( ℓ ) − g ( ℓ ) − A ( ℓ ) | . ( | g − | ∞ + | ℓ | β | g | β -Höl ) | ℓ | α | A | α -gr ,which proves (3.21). Lemma 3.33
Let α, β ∈ ( , ] such that β + α > and α + β > , A ∈ Ω α , and g ∈ G β . Then | A g − g − A | α ∧ β -vee . ( | g − | ∞ + | g | β -Höl ) | A | α , (3.23) where the proportionality constant depends only on α and β .Proof. Let ℓ, ¯ ℓ ∈ X form a vee. Recall the identity (3.22). By (3.7), (3.20), andYoung’s estimate (since β + α > ), we have (cid:12)(cid:12)(cid:12) Z (Ad ℓ g ( t ) − Ad ¯ ℓ g ( t ) ) d ℓ A ( t ) (cid:12)(cid:12)(cid:12) . | g | β -Höl Area ( ℓ, ¯ ℓ ) β/ | ℓ | α | A | α -gr . Similarly, using (3.7), (3.19), and Young’s estimate (since β + α > ), we have (cid:12)(cid:12)(cid:12) Z (Ad ¯ ℓ g ( t ) − ) d( ℓ A ( t ) − ¯ ℓ A ( t )) (cid:12)(cid:12)(cid:12) . ( | g − | ∞ + | g | β -Höl | ℓ | β ) | A | α Area ( ℓ, ¯ ℓ ) α/ . Note that the integrals on the left-hand sides of the previous two bounds add to ( A g − g − A )( ℓ ) − ( A g − g − A )( ¯ ℓ ) , from which (3.23) follows. onstruction of the state space Proof of Theorem 3.27.
The fact that the action of G β maps Ω α -gr into Ω α ∧ β -gr follows from Proposition 3.28 and Lemma 3.32. The fact that the action of G β maps Ω α into Ω α ∧ β follows by combining the equivalence of norms |·| α ≍ |·| α -gr + |·| α -vee from Theorem 3.11 with Proposition 3.28 and Lemmas 3.32 and 3.33. The fact that ( A, g ) A g is continuous in both cases follows from writing A g − B h = ( ( A − B ) h − h − ( A − B ) ) − ( ( A g ) hg − − hg − − A g ) − hg − + ( A − B ) and noting that all four terms vanish in Ω α -gr (resp. Ω α ) as ( A, g ) → ( B, h ) in Ω α -gr × G β (resp. Ω α × G β ) again by Proposition 3.28 and Lemmas 3.32 and 3.33.Finally, if α ≤ β , the fact that A gh = ( A h ) g follows from the identity d( gh ) ( gh ) − = (d g ) g − + Ad g [ (d h ) h − ] . Combining all of these claims completes the proof.
The main result of this subsection, Proposition 3.35, provides a way to recover thegauge transformation that transforms between gauge equivalent elements of Ω α -gr .This result can be seen as a version of [Sen92, Prop. 2.1.2] for the non-smooth case(see also [LN06, Lem. 3]).Let us fix α ∈ ( , ] throughout this subsection. For ℓ ∈ X and A ∈ Ω α -gr , theODE d y ( t ) = y ( t ) d ℓ A ( t ) , y ( ) = 1 ,admits a unique solution y : [ , ] → G as a Young integral (thanks to Lemma 3.14).Furthermore, the map ℓ A y is locally Lipschitz when both sides are equippedwith | · | α -Höl . We define the holonomy of A along ℓ as hol ( A, ℓ ) def = y ( ) . As usual,we extend the definition hol ( A, γ ) to any piecewise affine path γ : [ , ] → T bytaking the ordered product of the holonomies along individual line segments. Remark 3.34
Recall from item (i) of Theorem 2.1 that, provided α ∈ ( , ] and β ∈ ( α − , ] , the holonomy hol ( A, γ ) is well-defined for all paths γ piecewise in C ,β (rather than only piecewise affine) and all A ∈ Ω α .For any g ∈ G α and any piecewise affine path γ , note the familiar identityhol ( A g , γ ) = g ( γ ( )) hol ( A, γ ) g ( γ ( )) − . (3.24)For x, y ∈ T , let L xy denote the set of piecewise affine paths γ : [ , ] → T with γ ( ) = x and γ ( ) = y . Proposition 3.35
Let α ∈ ( , ] and A, ¯ A ∈ Ω α -gr . Then the following are equiv-alent:(i) A ∼ ¯ A . onstruction of the state space (ii) there exists x ∈ T and g ∈ G such that hol ( ¯ A, γ ) = g hol ( A, γ ) g − for all γ ∈ L xx .(iii) for every x ∈ T there exists g x ∈ G such that hol ( ¯ A, γ ) = g x hol ( A, γ ) g − x for all γ ∈ L xx .Furthermore, if (ii) holds, then there exists a unique g ∈ G α such that g ( x ) = g and A g = ¯ A . The element g is determined by g ( y ) = hol ( ¯ A, γ xy ) − g hol ( A, γ xy ) , (3.25) where γ xy is any element of L xy , and satisfies | g | α -Höl . | A | α -gr + | ¯ A | α -gr . (3.26) Proof.
The implication (i) ⇒ (iii) is clear from (3.24) and the implication (iii) ⇒ (ii)is trivial. Hence suppose (ii) holds. Let us define g ( y ) using (3.25), which we notedoes not depend on the choice of path γ xy ∈ L xy . Then one can readily verifythe bound (3.26) and that A g = ¯ A , which proves (i). The fact that g is the uniqueelement in G α such that g ( x ) = g and A g = ¯ A follows again from (3.24). We define and study in this subsection the space of gauge orbits of the Banach space Ω α . Let G ,α denote the closure of C ∞ ( T , G ) in G α . The following is a simplecorollary of Theorem 3.27. Corollary 3.36
Let α ∈ ( , ] . Then ( A, g ) A g is a continuous left group actionof G ,α on Ω α and on Ω α -gr .Proof. It holds that A g ∈ ı Ω C ∞ whenever A ∈ ı Ω C ∞ and g ∈ C ∞ ( T , G ) . Theconclusion follows from Theorem 3.27 by continuity of ( A, g ) A g .We are now ready to define our desired space of orbits. Definition 3.37
For α ∈ ( , ] , let O α denote the space of orbits Ω α / G ,α equipped with the quotient topology. For every A ∈ Ω α , let O α ∋ [ A ] def = { A g : g ∈ G ,α } ⊂ Ω α denote the corresponding gauge orbit.We next show that the restriction to the subgroup G ,α is natural in the sensethat G ,α is precisely the stabiliser of Ω α . For this, we use the following versionof the standard fact that the closure of smooth functions yield the “little Hölder”spaces. Lemma 3.38
For α ∈ ( , ) , one has g ∈ G ,α if and only if lim ε → sup | x − y | <ε | x − y | − α | g ( x ) − g ( y ) | = 0 . We then have the following general statement. onstruction of the state space
Let α ∈ ( , ) and A ∈ Ω α -gr . Suppose that A g ∈ Ω α -gr forsome g ∈ G α . Then g ∈ G ,α .Proof. By part (ii) of Proposition 3.25, lim ε → sup | ℓ | <ε | A ( ℓ ) | + | A g ( ℓ ) || ℓ | α = 0 . (3.27)Combining (3.27) with the expression for g in (3.25), we conclude that g ∈ G ,α by Lemma 3.38.In general, the quotient of a Polish space by the continuous action of a Polishgroup has no nice properties. In the remainder of this subsection, we show that thespace O α for α ∈ ( , ) is itself a Polish space and we exhibit a metric D α for itstopology. We first show that these orbits are very well-behaved in the followingsense. Lemma 3.40
Let α ∈ ( , ) . For every A ∈ Ω α , the gauge orbit [ A ] is closed.Proof. Since Ω α is a separable Banach space, it suffices to show that, for every B ∈ Ω α and any sequence A n ∈ [ B ] such that A n → A in Ω α , one has A ∈ [ B ] .Since the A n are uniformly bounded, the corresponding gauge transformations g n such that A n = B g n are uniformly bounded in G α by (3.26). Since G α ⊂ G β compactly for β < α , we can assume modulo passing to a subsequence that g n → g in G β , which implies that A = B g by Theorem 3.27. Since however we knowthat A ∈ Ω α , we conclude that g ∈ G ,α by Proposition 3.39, so that A ∈ [ B ] asrequired.In the next step, we introduce a complete metric k α on Ω α which generates thesame topology as | · | α , but shrinks distances at infinity so that, for large r , points onthe sphere with radius r are close to each other but such that the spheres with radii r and r are still far apart. We then define the metric D α on O α as the Hausdorffdistance associated with k α . Definition 3.41
Let α ∈ ( , ] . For A, B ∈ Ω α , set K α ( A, B ) def = || A | α − | B | α | + 1 ( | A | α ∧ | B | α ) + 1 ( | A − B | α ∧ ) ,and define the metric k α ( A, B ) def = inf Z ,...,Z n n X i =1 K α ( Z i − , Z i ) , (3.28)where the inf is over all finite sequences Z , . . . , Z n ∈ Ω α with Z = A and Z n = B . onstruction of the state space Note that k α ( A, B ) ≤ K α ( A, B ) ≤ r + 1 (3.29)for all A, B in the sphere S rα def = { C ∈ Ω α : | C | α = r } . On the other hand, for r , r > , if A ∈ S r α and B ∈ S r α , then K α ( A, B ) ≥ | r − r | r ∧ r + 1 , (3.30)and if r > and A, B are in the ball B rα def = { C ∈ Ω α : | C | α ≤ r } ,then K α ( A, B ) ≥ | A − B | α ∧ r + 1 . (3.31) Lemma 3.42
Let α ∈ ( , ] . If A ∈ S rα and B ∈ S r + hα for some r, h > , then k α ( A, B ) ≥ hr + h + 1 . Proof.
Consider a sequence Z = A, Z , . . . , Z n = B . Let r i def = | Z i | α and R def = max i =0 ,...,n r i . Then n X i =1 K α ( Z i − , Z i ) ≥ R − rR + 1 ≥ hr + h + 1 ,where in the first bound we used (3.30) and R − r ≤ P ni =1 | r i − − r i | , and in thesecond bound we used h ≤ R − r and that ≤ λ λr + λ +1 is increasing. Proposition 3.43
Let α ∈ ( , ] . The metric space ( Ω α , k α ) is complete and k α metrises the original topology of Ω α .Proof. It is obvious that k α is weaker than the metric induced by | · | α . On theother hand, if A n is a k α -Cauchy sequence, then sup | A n | α < ∞ by Lemma 3.42.It readily follows from (3.31) that k α ( A n , A ) → and | A − A n | α → for some A ∈ Ω α . Definition 3.44
Let α ∈ ( , ] . We denote by D α the Hausdorff distance on O α associated with k α . Theorem 3.45
The metric space ( O α , D α ) is complete and D α metrises the quo-tient topology on O α . In particular, O α is a Polish space. tochastic heat equation For the proof of Theorem 3.45 we require several lemmas.
Lemma 3.46
Let α ∈ ( , ] and x ∈ O α . Then for all r > inf A ∈ x | A | α thereexists A ∈ x with | A | α = r .Proof. For any A ∈ Ω α , from the identity (3.24), we can readily construct acontinuous function g : [ , ∞ ) → G ,α such that g ( ) ≡ and lim t →∞ | A g ( t ) | α = ∞ . The conclusion follows by continuity of g
7→ | A g | α (Corollary 3.36). Lemma 3.47
Suppose α ∈ ( , ] and that k α ( A n , A ) → . Then D α ([ A ] , [ A n ]) → .Proof. Consider ε > . Observe that (3.29), the fact that sup n | A n | α < ∞ byLemma 3.42, and Lemma 3.46 together imply that there exists r > sufficientlylarge such that the Hausdorff distance for k α between [ A n ] ∩ ( Ω α \ B rα ) and [ A ] ∩ ( Ω α \ B rα ) is at most ε for all n sufficiently large. On the other hand,for any r > , g ∈ G ,α , and X, Y ∈ Ω α such that X, X g ∈ B rα , it follows fromLemmas 3.32 and 3.33 and the identity X g − Y g = ( ( X − Y ) g − g − ( X − Y ) ) + ( X − Y ) that | X g − Y g | α . ( | g | α -Höl ) | X − Y | α where | g | α -Höl . r due to (3.26). It follows that sup X ∈ [ A n ] ∩ B rα inf Y ∈ [ A ] k α ( X, Y ) + sup X ∈ [ A ] ∩ B rα inf Y ∈ [ A n ] k α ( X, Y ) → ,which concludes the proof. Lemma 3.48
Let α ∈ ( , ] and suppose that [ A n ] is a D α -Cauchy sequence.Then there exist B ∈ Ω α and representatives B n ∈ [ A n ] such that k α ( B n , B ) → .Proof. We can assume that A n are “almost minimal” representatives of [ A n ] in thesense that | A n | α ≤ inf g ∈ G ,α | A gn | α . By Lemma 3.42 and the definition of theHausdorff distance, we see that sup n ≥ | A n | α < ∞ , from which it is easy to extracta k α -Cauchy sequence B n ∈ [ A n ] . Proof of Theorem 3.45.
Since every x ∈ O α is closed by Lemma 3.40, D α ( x, y ) =0 if and only if x = y . The facts that D α is a complete metric and that it metrisesthe quotient topology both follow from Lemmas 3.47 and 3.48. We investigate in this section the regularity of the stochastic heat equation (whichis the “rough part” of the SYM) with respect to the spaces introduced in Section 3.For the remainder of the article, we will focus on the space of “ -forms” Ω α . tochastic heat equation The main result of this subsection, Proposition 4.1, provides a convenient way toextend regularising properties of an operator K to the spaces Ω α . This will beparticularly helpful in deriving Schauder estimates and controlling the effect ofmollifiers (Corollaries 4.2 and 4.4).Let E be a Banach space throughout this subsection, and consider a linear map K : C ∞ ( T , E ) → C ( T , E ) . We denote also by K the linear map K : Ω C ∞ → Ω C obtained by componentwise extension. We denote by ˆ K : ı Ω C ∞ → ı Ω C the natural“lift” of K given by ˆ K ( ıA ) def = ı ( KA ) . We say that K is translation invariant if K commutes with all translation operators T v : f f ( · + v ) . For θ ≥ , we denote | K | C θ → L ∞ def = sup {| K ( f ) | ∞ | f ∈ C ∞ ( T , E ) , | f | C θ = 1 } . In general, for normed spaces
X, Y and a linear map K : D ( K ) → Y , where D ( K ) ⊂ X , we denote | K | X → Y def = sup {| K ( x ) | Y | x ∈ D ( K ) , | x | X = 1 } . If D ( K ) is dense in X , then K does of course extend uniquely to all of X if | K | X → Y < ∞ . The reason for this setting is that it will be convenient to consider D ( K ) as fixed and to allow X to vary. Proposition 4.1
Let < ¯ α ≤ α ≤ . Let K : C ∞ ( T , E ) → C ( T , E ) be atranslation invariant linear map. Then | ˆ K | Ω α → Ω α . | K | C ( α − ¯ α ) / → L ∞ . (4.1) Furthermore, if ¯ α ∈ [ α , α ] , then for all A ∈ ı Ω C ∞ | ˆ KA | ¯ α -gr . | K | C α − ¯ α → L ∞ | A | ( α − ¯ α ) /αα | A | ( α − α ) /αα -gr . (4.2) The proportionality constants in both inequalities are universal.Proof.
We suppose that | K | C α − ¯ α → L ∞ < ∞ , as otherwise there is nothing to prove.Let A ∈ Ω C ∞ and observe that, for ( x, v ) ∈ X , Z ( KA i )( x + tv ) d t = Z T tv ( KA i )( x ) d t = Z K [ T tv A i ]( x ) d t = K h Z T tv A i d t i ( x ) = K h Z A i ( · + tv ) d t i ( x ) ,where we used translation invariance of K in the second equality, and the bound-edness of K in the third equality. In particular, it follows from the definition of ı : Ω C → Ω that ˆ K ( ıA )( x, v ) = K [ ıA ( · , v )]( x ) . (4.3) tochastic heat equation We will first prove (4.2). We claim that for any θ ∈ [ , ] | ıA ( · , v ) | C ( θα/ ) . | ıA | θα | ıA | − θα -gr | v | α ( − θ/ ) (4.4)for a universal proportionality constant. Indeed, note that | ıA ( x, v ) | ∞ ≤ | ıA | α -gr | v | α which is bounded above by the right-hand side of (4.4) for any θ ∈ [ , ] . Further-more, we have for all x, y ∈ T | ıA ( x, v ) − ıA ( y, v ) | . [ | ıA | α | v | α/ | x − y | α/ ] ∧ [ | ıA | α -gr | v | α ] , (4.5)for a universal proportionality constant, from which (4.4) follows by interpolation.If ¯ α ∈ [ α , α ] , then we can take θ = 2 ( α − ¯ α ) /α in (4.4) and combine with (4.3) toobtain (4.2).We now prove (4.1). Consider ℓ = ( x, v ) , ¯ ℓ = ( ¯ x, ¯ v ) ∈ X . If ℓ, ¯ ℓ are far, thenthe necessary estimate follows from (4.2). Hence, suppose ℓ, ¯ ℓ are not far. Considerthe function Ψ ∈ C ∞ ( T , E ) given by Ψ ( y ) def = ıA ( y, v ) − ıA ( y + ¯ x − x, ¯ v ) . Notethat (4.3) implies ( K Ψ )( x ) = ˆ K ( ıA )( ℓ ) − ˆ K ( ıA )( ¯ ℓ ) . (4.6)We claim that for any θ ∈ [ , ] | Ψ | C αθ/ . | A | α | v | αθ/ ̺ ( ℓ, ¯ ℓ ) α ( − θ ) . (4.7)Indeed, note that | Ψ | ∞ ≤ | A | α ̺ ( ℓ, ¯ ℓ ) α , which is bounded above (up to a universalconstant) by the right-hand side of (4.7) for any θ ∈ [ , ] . Furthermore, since ( y, v ) , ( y + ¯ x − x, ¯ v ) are also not far for every y ∈ T , and since | v | ≍ | ¯ v | , we have | Ψ ( y ) − Ψ ( z ) | = | A ( y, v ) − A ( y + ¯ x − x, ¯ v ) − A ( z, v ) + A ( z + ¯ x − x, ¯ v ) | . | A | α [ ̺ ( ℓ, ¯ ℓ ) α ∧ ( | v | α/ | y − z | α/ )] (4.8)for a universal proportionality constant, from which (4.7) follows by interpolation.Taking θ = α − ¯ αα ⇔ ¯ α = α ( − θ ) in (4.7) and combining with (4.6) proves (4.1).As a consequence of Proposition 4.1, any linear map K : C ∞ ( T , E ) → C ( T , E ) with | K | C ( α − ¯ α ) / → L ∞ < ∞ uniquely determines a bounded linear map ˆ K : Ω α → Ω α which intertwines with K through the embedding ı : Ω C → Ω . The sameapplies to Ω α -gr → Ω α -gr if | K | L ∞ → L ∞ < ∞ . In the sequel, we will denote ˆ K bythe same symbol K without further notice.We give two useful corollaries of Proposition 4.1. For t ≥ , let e t ∆ denote theheat semigroup acting on C ∞ ( T , E ) . Corollary 4.2
Let < ¯ α ≤ α ≤ . Then for all A ∈ Ω α , it holds that | ( e t ∆ − ) A | ¯ α . t ( α − ¯ α ) / | A | α , (4.9) where the proportionality constant depends only on α − ¯ α . tochastic heat equation Proof.
Recall the classical estimate for κ ∈ [ , ] | ( e t ∆ − ) | C κ → L ∞ ≤ | ( e t ∆ − ) | C κ -Höl → L ∞ . t κ/ . The claim then follows from (4.1) by taking κ = ( α − ¯ α ) / . Remark 4.3
The appearance of t ( α − ¯ α ) / in (4.9) may seem unusual since oneinstead has t ( α − ¯ α ) / in the classical Schauder estimates for the Hölder norm | · | C α .The exponent ( α − ¯ α ) / is however sharp (which can be seen by looking at theFourier basis), and is consistent with the embedding of C α/ into Ω α (Remark 3.23). Corollary 4.4
Let < ¯ α ≤ α ≤ and κ ∈ [ , ] . Let χ be a mollifier on R × R and consider a function A : R → Ω α . Then for any interval I ⊂ R sup t ∈ I | ( χ ε ∗ A )( t ) − A ( t ) | ¯ α . | χ | L (cid:16) ε ( α − ¯ α ) / sup t ∈ I ε | A ( t ) | α + ε κ | A | C κ -Höl ( I ε , Ω ¯ α ) (cid:17) ,where I ε is the ε fattening of I , and the proportionality constant is universal.Proof. For t ∈ R define m ( t ) def = R T χ ε ( t, x ) d x and denote by χ ε ( t ) the convolutionoperator [ χ ε ( t ) f ]( x ) def = h χ ε ( t, x − · ) , f ( · ) i for f ∈ D ′ ( T ) . Observe that for any θ ∈ [ , ] | m ( t ) f − χ ε ( t ) f | L ∞ ≤ ε θ | χ ε ( t, · ) | L ( T ) | f | C θ -Höl ( T ) . In particular, | m ( t ) − χ ε ( t ) | C ( α − ¯ α ) / -Höl → L ∞ ≤ ε ( α − ¯ α ) / | χ ε ( t, · ) | L ( T ) . Hence, forany t ∈ I , | ( χ ε ∗ A )( t ) − A ( t ) | ¯ α ≤ Z R | ( χ ε ( s ) − m ( s )) A ( t − s ) | ¯ α d s + Z R | m ( s )( A ( t − s ) − A ( t )) | ¯ α d s . Z R | χ ε ( s, · ) | L ε ( α − ¯ α ) / | A ( t − s ) | α d s + Z R | m ( s ) || s | κ | A | C κ -Höl ( I ε , Ω ¯ α ) d s ≤ | χ | L ε ( α − ¯ α ) / sup t ∈ I ε | A ( t ) | α + ε κ | χ | L | A | C κ -Höl ( I ε , Ω ¯ α ) ,where we used (4.1) in the second inequality.Another useful property is that the heat semigroup is strongly continuous on Ω α . To show this, we need the following lemma. For a function ω : R + → R + , let C ω ( T , E ) denote the space of continuous functions f : T → E with | f | C ω def = sup x = y | f ( x ) − f ( y ) | ω ( | x − y | ) < ∞ . tochastic heat equation Let α ∈ ( , ] and K : C ∞ ( T , E ) → C ( T , E ) be a translationinvariant linear map with | K | L ∞ → L ∞ < ∞ . Let A ∈ Ω α and ω : R + → R + , andsuppose that for all x, y ∈ T , v, ¯ v ∈ B / , and h ∈ R , | A ( x, v ) − A ( y, v ) | ≤ | A | α -gr | v | α ω ( | x − y | ) (4.10) and | A ( x, v ) − A ( x + h, ¯ v ) − A ( y, v ) + A ( y + h, ¯ v ) | ≤ | A | α ̺ ( ℓ, ¯ ℓ ) α ω ( | x − y | ) , (4.11) where ℓ = ( x, v ) and ¯ ℓ = ( x + h, ¯ v ) . Then | KA | α -gr ≤ | K | C ω → L ∞ | A | α -gr and | KA | α ≤ | K | C ω → L ∞ | A | α . Proof.
The proof is essentially the same as that of Proposition 4.1; one simplyreplaces (4.5) by (4.10) and (4.8) by (4.11).
Proposition 4.6
Let α ∈ ( , ] . The heat semigroup e t ∆ is strongly continuous on Ω α -gr and Ω α .Proof. Observe that for every A ∈ ı Ω C ∞ there exists a bounded modulus ofcontinuity ω : R + → R + such that (4.10) and (4.11) hold. On the other hand, recallthat for every bounded modulus of continuity ω : R + → R + lim t → | e t ∆ − | C ω → L ∞ = 0 . It follows from Lemma 4.5 that lim t → | e t ∆ A − A | α -gr = 0 for every A ∈ Ω C ∞ ,and the same for the norm | · | α , from which the conclusion follows by density of ı Ω C ∞ in Ω α -gr and Ω α . In this subsection, let ξ be a g -valued Gaussian random distribution on R × T . Weassume that there exists C ξ > such that E [ |h ξ, ϕ i| ] ≤ C ξ | ϕ | L ( R × T ) (4.12)for all smooth compactly supported ϕ : R × T → R . Let ξ , ξ be two i.i.d. copiesof ξ , and let Ψ = P i =1 Ψ i d x i solve the stochastic heat equation ( ∂ t − ∆ ) Ψ = ξ on R + × T with initial condition ι Ψ ( ) ∈ Ω α . Lemma 4.7
Let P be a triangle with inradius h . Let κ ∈ ( , ) and p ∈ [ , κ − ) ,and let W κ,p denote the Sobolev–Slobodeckij space on T . Then | ˚ P | pW κ,p . | P | h − κp , where the proportionality constant depends only on κp . tochastic heat equation Proof.
Using the definition of Sobolev–Slobodeckij spaces, we have | ˚ P | pW κ,p = Z T Z T | ˚ P ( x ) − ˚ P ( y ) | p | x − y | κp +2 d x d y = 2 Z ˚ P Z T \ ˚ P | x − y | − κp − d x d y ≤ Z ˚ P d x Z | y − x | >d ( x,∂P ) d y | x − y | − κp − . Z ˚ P d ( x, ∂P ) − κp d x = Z |{ x ∈ ˚ P | d ( x, ∂P ) < δ }| δ − − κp d δ . Note that the integrand is non-zero only if δ < h , in which case |{ x ∈ ˚ P | d ( x, ∂P ) < δ }| . | ∂P | δ , where | ∂P | denotes the length of the perimeter. Hence,since κp < , | ˚ P | pW κ,p . Z h | ∂P | δ − κp d δ . | ∂P | h − κp +1 ≍ | P | h − κp ,as claimed. Lemma 4.8
Let κ ∈ ( , ) and suppose Ψ ( ) = 0 . Then for any triangle P withinradius h E [ | Ψ ( t )( ∂P ) | ] . C ξ t κ | P | h − κ ≤ C ξ t κ | P | − κ ,where the proportionality constant depends only on κ .Proof. By Stokes’ theorem, we have | Ψ ( t )( ∂P ) | = |h ∂ Ψ ( t ) − ∂ Ψ ( t ) , ˚ P i| . Observe that h ∂ Ψ ( t ) , ˚ P i = Z R × T ξ ( s, y ) s ∈ [ ,t ] [ e ( t − s ) ∆ ∂ ˚ P ]( y ) d s d y . Hence, by (4.12), E [ |h ∂ Ψ ( t ) , ˚ P i| ] ≤ C ξ Z t | e s ∆ ∂ ˚ P | L d s . By the estimate | e s ∆ f | L . s ( − κ ) / | f | H − κ , we have | e s ∆ ∂ ˚ P | L . s − κ | ∂ ˚ P | H − κ . Since | ∂ f | H − κ . | f | H κ , we have by Lemma 4.7 E [ |h ∂ Ψ ( t ) , ˚ P i| ] . C ξ Z s − κ | P | h − κ d s . C ξ t κ | P | h − κ . Likewise for the term h ∂ Ψ ( t ) , ˚ P i , and the conclusion follows from the inequality πh ≤ | P | . tochastic heat equation Let ℓ = ( x, v ) ∈ X and consider the distribution h δ ℓ , ψ i def = R | v | ψ ( x + tv ) d t . Then, for any κ ∈ ( , ) , | δ ℓ | H − κ . | ℓ | κ ,where the proportionality constant depends only on κ .Proof. By rotation and translation invariance, we may assume ℓ = ( , | ℓ | e ) . For k = ( k , k ) ∈ Z , we have h δ ℓ , e πi h k, ·i i = ( e πik | ℓ | − ) / ( πik ) . Hence | δ ℓ | H − κ = X k ∈ Z |h δ ℓ , e πi h k, ·i i| ( k + k ) − κ . X k ∈ Z ( | ℓ | ∧ k − )( k + k ) − κ . X k ∈ Z ( | ℓ | ∧ k − )( k ) − κ . Splitting the final sum into | k | ≤ | ℓ | − and | k | > | ℓ | − yields the desired result. Lemma 4.10
Let κ ∈ ( , ) and suppose Ψ ( ) = 0 . Then for any ℓ ∈ X E [ | Ψ ( t )( ℓ ) | ] . C ξ t κ | ℓ | − κ ,where the proportionality constant depends only on κ .Proof. Observe that Ψ ( t )( ℓ ) = P i =1 | v | − v i h Ψ i ( t ) , δ ℓ i , where we used the notationof Lemma 4.9. Furthermore, h Ψ i ( t ) , δ ℓ i = Z R × T ξ i ( s, y ) s ∈ [ ,t ] [ e ( t − s ) ∆ δ ℓ ]( y ) d s d y . Hence, by (4.12), E [ h Ψ i ( t ) , δ ℓ i ] ≤ C ξ Z t | e s ∆ δ ℓ | L d s . The estimate | e s ∆ f | L . s ( κ − ) / | f | H κ − implies | e s ∆ δ ℓ | L . s κ − | δ ℓ | H κ − .Hence, by Lemma 4.9, E [ h Ψ i ( t ) , δ ℓ i ] . C ξ Z s − κ | ℓ | − κ d s . C ξ t κ | ℓ | − κ ,and the claim follows from the bound || v | − v i | . .Since our “index space” X and “distance” function ̺ are not entirely standard,we spell out the following Kolmogorov-type criterion. tochastic heat equation Let A be a g -valued stochastic process indexed by X such that, forall joinable ℓ, ¯ ℓ ∈ X , A ( ℓ ⊔ ¯ ℓ ) = A ( ℓ ) + A ( ¯ ℓ ) almost surely. Suppose that thereexist p ≥ , M > , and α ∈ ( , ] such that for all ℓ ∈ X E [ | A ( ℓ ) | p ] ≤ M | ℓ | pα ,and for all triangles P E [ | A ( ∂P ) | p ] ≤ M | P | pα/ . Then there exists a modification of A (which we denote by the same letter) whichis a.s. a continuous function on X . Furthermore, for every ¯ α ∈ ( , α − p ) , thereexists λ > , depending only on p, α, ¯ α , such that E [ | A | p ¯ α ] ≤ λM . Proof.
Observe that for any ℓ, ¯ ℓ ∈ X , we can write A ( ℓ ) − A ( ¯ ℓ ) = A ( ∂P ) + A ( ∂P ) + A ( a ) − A ( b ) , where | P | + | P | ≤ ̺ ( ℓ, ¯ ℓ ) and | a | + | b | ≤ ̺ ( ℓ, ¯ ℓ ) (if ℓ, ¯ ℓ are far, then a = ℓ , b = ¯ ℓ , and P , P are empty). It follows that for all ℓ, ¯ ℓ ∈ X E [ | A ( ℓ ) − A ( ¯ ℓ ) | p ] . M ̺ ( ℓ, ¯ ℓ ) pα , (4.13)where the proportionality constant depends only on p, α . For N ≥ let D N denote the set of line segments in X whose start and end points have dyadiccoordinates of scale − N , and let D = ∪ N ≥ D N . For r > , ℓ ∈ X , define B ̺ ( r, ℓ ) def = { ¯ ℓ ∈ X | ̺ ( ¯ ℓ, ℓ ) ≤ r } . From Definition 3.3 and Remark 3.4, we seethat for some K > , the family { B ̺ ( K − N , ℓ ) } ℓ ∈ D N covers X (quite wastefully)for every N ≥ . It readily follows, using (3.1), that for any ¯ α ∈ ( , ]sup ℓ, ¯ ℓ ∈ D | A ( ℓ ) − A ( ¯ ℓ ) | p ̺ ( ℓ, ¯ ℓ ) ¯ αp . X N ≥ X a,b ∈ D N ̺ ( a,b ) ≤ K − N N ¯ αp | A ( a ) − A ( b ) | p . (4.14)Observe that | D N | ≤ N , and thus the second sum has at most N terms. Hence,for ¯ α ∈ ( , α − p ) ⇔
16 + p ( ¯ α − α ) < , we see from (4.13) that the expectationof the right-hand side of (4.14) is bounded by λ ( p, α, ¯ α ) M . The conclusion readilyfollows as in the classical Kolmogorov continuity theorem.By equivalence of moments for Gaussian random variables, Lemmas 4.8, 4.10,and 4.11 yield the following lemma. Lemma 4.12
Suppose Ψ ( ) = 0 . Then for any p > , α ∈ ( , ) , and ¯ α ∈ ( , α − p ) , there exists C > , depending only on p, ¯ α, α , such that for all t ≥ E [ | Ψ ( t ) | p ¯ α ] ≤ CC p/ ξ t p ( − α ) / . We are now ready to prove the following continuity theorem. tochastic heat equation
Let < ¯ α < α < , κ ∈ ( , α − ¯ α ) , and suppose Ψ ( ) ∈ Ω α . Thenfor all p ≥ and any T > E h sup ≤ s
Corollary 4.14
Let χ be a mollifier on R × T . Suppose that ξ is a g -valuedwhite noise and denote ξ ε def = χ ε ∗ ξ . Suppose that Ψ ( ) = 0 and let Ψ ε solve ( ∂ t − ∆ ) Ψ ε = ξ ε on R + × T with zero initial condition Ψ ε ( ) = 0 . Let α ∈ ( , ) , T > , κ ∈ ( , − α ) , and p ≥ . Then E h sup t ∈ [ ,T ] | Ψ ε ( t ) − Ψ ( t ) | pα i /p . ε κ | χ | L ,where the proportionality constant depends only on α, κ, T, p .Proof. Observe that, by Theorem 4.13, E h sup t ∈ [ ,ε ] | Ψ ( t ) | pα i /p . ε κ . (4.15)Furthermore, for any ϕ ∈ L ( R × T ) , by Young’s inequality, E [ h ξ ε , ϕ i ] = | χ ε ∗ ϕ | L ≤ | ϕ | L | χ ε | L . tochastic heat equation Hence ξ ε satisfies (4.12) with C ξ ε def = | χ | L . It follows again by Theorem 4.13 that E h sup t ∈ [ ,ε ] | Ψ ε ( t ) | pα i /p . ε κ | χ | L . (4.16)It remains to estimate E [sup t ∈ [ ε ,T ] | Ψ ( t ) − Ψ ε ( t ) | pα ] .Denoting I def = [ ε , T ] , observe that by Corollary 4.4, for any ¯ α ∈ [ α, ] E h sup t ∈ I | Ψ ( t ) − χ ε ∗ Ψ ( t ) | pα i /p . | χ | L n ε ( ¯ α − α ) / E h sup t ∈ I ε | Ψ ( t ) | p ¯ α i /p + ε κ E h | Ψ | p C κ -Höl ( I ε , Ω α ) i /p o . Both expectations are finite provided ¯ α < , and thus the right-hand side is boundedabove by a multiple of ε κ | χ | L .We now estimate E [sup t ∈ I | χ ε ∗ Ψ − Ψ ε | pα ] . Let us denote by + the indicator onthe set { ( t, x ) ∈ R × T | t ≥ } . Observe that + ( χ ε ∗ ξ )( t, x ) and χ ε ∗ ( + ξ )( t, x ) both vanish if t < − ε and agree if t > ε . In particular, χ ε ∗ Ψ and Ψ ε both solve the (inhomogeneous) heat equation on [ ε , ∞ ) × T with the samesource term but with possibly different initial conditions. To estimate these initialconditions, for s ∈ [ − ε , ε ] , let us denote by χ ε ( s ) the convolution operator [ χ ε ( s ) f ]( x ) def = h χ ε ( s, x − · ) , f ( · ) i for f ∈ D ′ ( T ) . Observe that | χ ε ( s ) | L ∞ → L ∞ ≤ µ ( s ) def = Z T | χ ε ( s, x ) | d x ,and thus | χ ε ( s ) A | α . µ ( s ) | A | α for any A ∈ Ω α by (4.1). Hence E [ | χ ε ∗ Ψ ( ε ) | pα ] /p = E h(cid:12)(cid:12)(cid:12) Z R χ ε ( s ) Ψ ( ε − s ) d s (cid:12)(cid:12)(cid:12) pα i /p . | χ | L E h sup t ∈ [ , ε ] | Ψ | pα i /p . | χ | L ε κ . As a result, by Theorem 4.13 and recalling that ξ ε satisfies (4.12) with C ξ ε = | χ | L ,we obtain E [ | χ ε ∗ Ψ ( ε ) | pα ] /p + E [ | Ψ ε ( ε ) | pα ] /p . ε κ | χ | L . Finally, E [ sup t ≥ ε | χ ε ∗ Ψ ( t ) − Ψ ε ( t ) | pα ] /p . E [ | χ ε ∗ Ψ ( ε ) − Ψ ε ( ε ) | pα ] /p . ε κ | χ | L where we used Corollary 4.2 in the first inequality. egularity structures for vector-valued noises As already mentioned in the introduction, the aim of this section is to provide asolution / renormalisation theory for SPDEs of the form ( ∂ t − L t ) A t = F t ( A , ξ ) , t ∈ L + , (5.1)where the nonlinearities ( F t ) t ∈ L + , linear operators ( L t ) t ∈ L + , and noises ( ξ t ) t ∈ L − ,satisfy the assumptions required for the general theory of [Hai14, CH16, BCCH17,BHZ19] to apply. The problem is that this theory assumes that the differentcomponents of the solutions A t and of its driving noises ξ t are scalar-valued. Whilethis is not a restriction in principle (simply expand solutions and noises accordingto some arbitrary basis of the corresponding spaces), it makes it rather unwieldyto obtain an expression for the precise form of the counterterms generated by therenormalisation procedure described in [BCCH17].Instead, one would much prefer a formalism in which the vector-valued na-tures of both the solutions and the driving noises are preserved. To motivate ourconstruction, consider the example of a g -valued noise ξ , where g is some finite-dimensional vector space. One way of describing it in the context of [Hai14] wouldbe to choose a basis { e , . . . , e n } of g and to consider a regularity structure T withbasis vectors Ξ i endowed with a model Π such that Π Ξ i = ξ i with ξ i such that ξ = P ni =1 ξ i e i . We could then also consider the element Ξ ∈ T ⊗ g obtainedby setting Ξ = P ni =1 Ξ i ⊗ e i . When applying the model to Ξ (or rather its firstfactor), we then obtain ΠΞ = P ni =1 ξ i e i = ξ as expected. A cleaner coordinate-independent way of achieving the same result is to view the subspace T [ Ξ ] ⊂ T spanned by the Ξ i as a copy of g ∗ , with Π given by Π Ξ g = g ( ξ ) for any element Ξ g in this copy of g ∗ . In this way, Ξ ∈ T [ Ξ ] ⊗ g ≃ g ∗ ⊗ g is simply given by Ξ = id g ,where id g denotes the identity map g → g , modulo the canonical correspondence L ( X, Y ) ≃ X ∗ ⊗ Y . (Here and below we will use the notation X ≃ Y to denotethe existence of a canonical isomorphism between objects X and Y .) Remark 5.1
This viewpoint is consistent with the natural correspondence betweena g -valued rough path X and a model Π . Indeed, while X takes values in H ∗ , thetensor series / Grossman–Larson algebra over g , one evaluates the model Π againstelements of its predual H , the tensor / Connes–Kreimer algebra over g ∗ , see [Hai14,Sec. 4.4] and [BCFP19, Sec. 6.2].Imagine now a situation in which we are given g and g -valued noises ξ = and ξ = , as well as an integration kernel K which we draw as a plain line,and consider the symbol . It seems natural in view of the above discussion toassociate it with a subspace of T isomorphic to g ∗ ⊗ g ∗ (let’s borrow the notationfrom [GH19] and denote this subspace as ⊗ g ∗ ⊗ g ∗ ) and to have the canonicalmodel act on it as Π ( ⊗ g ⊗ g ) = ( K ⋆ g ( ξ ) ) ( K ⋆ g ( ξ ) ) . egularity structures for vector-valued noises It would appear that such a construction necessarily breaks the commutativity ofthe product since in the same vein one would like to associate to a copy of g ∗ ⊗ g ∗ , but this can naturally be restored by simply postulating that in T one hasthe identity ( ⊗ g ) · ( ⊗ g ) = ⊗ g ⊗ g = ⊗ g ⊗ g = ( ⊗ g ) · ( ⊗ g ) . (5.2)This then forces us to associate to a copy of the symmetric tensor product g ∗ ⊗ s g ∗ .The goal of this section is to provide a functorial description of such considerationswhich allows us to transfer algebraic identities for regularity structures of trees ofthe type considered in [BHZ19] to the present setting where each noise (or edge)type t is associated to a vector space g ∗ t . This systematises previous constructionslike [GH19, Sec. 3.1] or [Sch18, Sec. 3.1] where similar considerations were madein a rather ad hoc manner. Our construction bears a resemblance to that of [CW16]who introduced a similar formalism in the context of rough paths, but our formalismis more functorial and better suited for our purposes. Fix a collection L of types and recall that a “typed set” T consists of a finite set(which we denote again by T ) together with a map t : T → L . For any two typedsets T and ¯ T , write Iso( T, ¯ T ) for the set of all type-preserving bijections from T → ¯ T . Definition 5.2 A symmetric set s consists of a non-empty index set A s , as well asa triple s = ( { T a s } a ∈ A s , { t a s } a ∈ A s , { Γ a,b s } a,b ∈ A s ) where ( T a s , t a s ) is a typed set and Γ a,b s ⊂ Iso( T b s , T a s ) are non-empty sets such that, for any a, b, c ∈ A s , γ ∈ Γ a,b s ⇒ γ − ∈ Γ b,a s , γ ∈ Γ a,b s , ¯ γ ∈ Γ b,c s ⇒ γ ◦ ¯ γ ∈ Γ a,c s . In other words, a symmetric set is a connected groupoid inside
Set L , the categoryof typed sets endowed with type-preserving maps. Remark 5.3
Each of the sets Γ a,a s forms a group and, by connectedness, these areall (not necessarily canonically) isomorphic. We will call this isomorphism classthe “local symmetry group” of s .Given a typed set T and a symmetric set s , we define Hom( T, s ) = (cid:16) [ a ∈ A s Iso(
T, T a s ) (cid:17). Γ s ,i.e. we postulate that Iso(
T, T a s ) ∋ ϕ ∼ ˜ ϕ ∈ Iso(
T, T b s ) if and only if there exists γ ∈ Γ a,b s such that ϕ = γ ◦ ˜ ϕ . Note that, by connectedness of Γ s , any equivalenceclass in Hom( T, s ) has, for every a ∈ A s , at least one representative ϕ a ∈ Iso(
T, T a s ) . egularity structures for vector-valued noises Given two symmetric sets s and ¯ s , we also define the set SHom( s , ¯ s ) of “sections”by SHom( s , ¯ s ) = { Φ = ( Φ a ) a ∈ A s : Φ a ∈ Vec ( Hom( T a s , ¯ s ) ) } ,where Vec( X ) denotes the real vector space spanned by a set X . We then have thefollowing definition. Definition 5.4 A morphism between two symmetric sets s and ¯ s is a Γ s -invariantsection; namely, an element of Hom( s , ¯ s ) = { Φ ∈ SHom( s , ¯ s ) : Φ b = Φ a ◦ γ a,b ∀ a, b ∈ A s , ∀ γ a,b ∈ Γ a,b s } . Here, we note that right composition with γ a,b gives a well-defined map from Hom( T b s , ¯ s ) to Hom( T a s , ¯ s ) and we extend this to Vec(Hom( T b s , ¯ s )) by linearity.Composition of morphisms is defined in the natural way by ( ¯Φ ◦ Φ ) a = ¯Φ ¯ a ◦ Φ ( ¯ a ) a ,where Φ ( ¯ a ) a denotes an arbitrary representative of Φ a in Vec ( Iso( T a s , T ¯ a ¯ s ) ) and com-position is extended bilinearly. It is straightforward to verify that this is indepen-dent of the choice of ¯ a and of representative Φ ( ¯ a ) a thanks to the invariance property ¯Φ ¯ a ◦ γ ¯ a, ¯ b = ¯Φ ¯ b , as well as the postulation of the equivalence relation in the definitionof Hom( T a s , ¯ s ) . Remark 5.5
A natural generalisation of this construction is obtained by replacing L by an arbitrary finite category. In this case, typed sets are defined as before, witheach element having as type an object of L . Morphisms between typed sets A and ¯ A are then given by maps ϕ : A → ¯ A × Hom L such that, writing ϕ = ( ϕ , ~ϕ ) , onehas ~ϕ ( a ) ∈ Hom L ( t ( a ) , t ( ϕ ( a ))) for every a ∈ A . Composition is defined in theobvious way by “following the arrows”, namely ( ψ ◦ ϕ ) = ψ ◦ ϕ , ( ψ ◦ ϕ )( a ) = ~ψ ( ϕ ( a )) ◦ ~ϕ ( a ) ,where the composition on the right takes place in Hom L . The set Iso( A, ¯ A ) is thendefined as those morphisms ϕ such that ϕ is a bijection, but we do not impose that ~ϕ ( a ) is an isomorphism in L for a ∈ A . Remark 5.6
Note that, for any symmetric set s , there is a natural identity element id s ∈ Hom( s , s ) given by a [id T a s ] , with [id T a s ] denoting the equivalence classof id T a s in Hom( T a s , s ) . In particular, symmetric sets form a category, which wedenote by SSet (or
SSet L ). Remark 5.7
We choose to consider formal linear combinations in our definitionof
SHom since otherwise the resulting definition of
Hom( s , ¯ s ) would be too smallfor our purpose. egularity structures for vector-valued noises An important special case is given by the case when A s and A ¯ s aresingletons. In this case, Hom( s , ¯ s ) can be viewed as a subspace of Vec ( Hom( T s , ¯ s ) ) , Hom( T s , ¯ s ) = Iso( T s , T ¯ s ) / Γ ¯ s , and Γ ¯ s is a subgroup of Iso( T ¯ s , T ¯ s ) . Remark 5.9
An alternative, more symmetric, way of viewing morphisms of
SSet is as two-parameter maps A s × A ¯ s ∋ ( a, ¯ a ) Φ ¯ a,a ∈ Vec ( Iso( T a s , T ¯ a ¯ s ) ) ,which are invariant in the sense that, for any γ a,b ∈ Γ a,b s and ¯ γ ¯ a, ¯ b ∈ Γ ¯ a, ¯ b ¯ s , one hasthe identity Φ ¯ a,a ◦ γ a,b = ¯ γ ¯ a, ¯ b ◦ Φ ¯ b,b . (5.3)Composition is then given by ( ¯Φ ◦ Φ ) ¯¯ a,a = ¯Φ ¯¯ a, ¯ a ◦ Φ ¯ a,a ,for any fixed choice of ¯ a (no summation). Indeed, it is easy to see that for anychoice of ¯ a , ¯Φ ◦ Φ satisfies (5.3). To see that our definition does not depend on thechoice of ¯ a , note that, for any ¯ b ∈ A ¯ s , we can take an element γ ¯ a, ¯ b ∈ Γ ¯ a, ¯ b s (which isnon-empty set by definition) and use (5.3) to write id T ¯¯ a s ,T ¯¯ a s ◦ ¯Φ ¯¯ a, ¯ b ◦ Φ ¯ b,a = ¯Φ ¯¯ a, ¯ a ◦ ¯ γ ¯ a, ¯ b ◦ Φ ¯ b,a = ¯Φ ¯¯ a, ¯ a ◦ Φ ¯ a,a ◦ id T a s ,T a s . We write
Hom ( s , ¯ s ) of the set of morphisms, as described above, between s and ¯ s . To see that this notion of morphism gives an equivalent category note that themap(s) ι s , ¯ s : Hom( s , ¯ s ) → Hom ( s , ¯ s ) , given by mapping Γ s equivalence classesto their symmetrised sums, is a bijection and maps compositions in Hom to thecorresponding compositions in
Hom . Remark 5.10
The category
SSet of symmetric sets just described is an R -linearsymmetric monoidal category, with tensor product s ⊗ ¯ s given by A = A s × A ¯ s , T ( a, ¯ a ) = T a s ⊔ T ¯ a ¯ s , t ( a, ¯ a ) = t a s ⊔ t ¯ a ¯ s , Γ ( a, ¯ a ) , ( b, ¯ b ) = { γ ⊔ ¯ γ : γ ∈ Γ a,b s , ¯ γ ∈ Γ ¯ a, ¯ b ¯ s } . and unit object given by A = {•} a singleton and T • = . Remark 5.11
We will sometimes encounter the situation where a pair ( s , ¯ s ) ofsymmetric sets naturally comes with elements Φ a ∈ Hom( T a s , ¯ s ) such that Φ ∈ Hom( s , ¯ s ) . In this case, Φ is necessarily an isomorphism which we call the “canon-ical isomorphism” between s and ¯ s . Note that this notion of “canonical” is notintrinsic to SSet but relies on additional structure in general.More precisely, consider a category C that is concrete over typed sets (i.e. suchthat objects of C can be viewed as typed sets and morphisms as type-preserving mapsbetween them). Then, any collection ( T a ) a ∈ A of isomorphic objects of C yields a egularity structures for vector-valued noises symmetric set s by taking for Γ the groupoid of all C -isomorphisms between them.Two symmetric sets obtained in this way such that the corresponding collections ( T a ) a ∈ A and ( ¯ T b ) b ∈ ¯ A consist of objects that are C -isomorphic are then canonicallyisomorphic (in SSet ) by taking for Φ a the set of all C -isomorphisms from T a toany of the ¯ T b . Note that this does not in general mean that there isn’t anotherisomorphism between these objects in SSet ! Example 5.12
An example of a symmetric set is obtained via “a tree with L -typedleaves”. A concrete tree τ is defined by fixing a vertex set V , which we can takewithout loss of generality as a finite subset of N (which we choose to play the roleof the set of all possible vertices), together with an (oriented) edge set E ⊂ V × V so that the resulting graph is a rooted tree with all edges oriented towards the root,as well as a labelling t : L → L , with L ⊂ V the set of leaves. However, whenwe draw a“tree with L -typed leaves” such as , we are actually specifying aisomorphism class of trees since we are not specifying V , E , and t as concrete sets.Thus corresponds to an infinite isomorphism class of trees τ with each τ ∈ τ being a concrete representative of τ .For any two concrete representatives τ = ( V , E , t ) and τ = ( V , E , t ) inthe isomorphism class , we have two distinct tree isomorphisms γ : V ⊔ E → V ⊔ E which preserve the typed tree structure, since it doesn’t matter how the twovertices of type get mapped onto each other. In this way, we have an unambiguousway of viewing τ = as an object in h τ i ∈ SSet , with typed set ( L, t ) and localsymmetry group isomorphic to Z . Example 5.13
We now give an example where we compute
Hom( • , • ) and Hom( • , • ) .Consider τ = , fix some representative τ ∈ τ , and write T τ = { x, y, z } ⊂ V τ ⊂ N , with t τ ( x ) = t τ ( y ) = and t τ ( z ) = – the local symmetry group isthen isomorphic to Z , acting on T τ by permuting { x, y } . We also introduce asecond isomorphism class ¯ τ = which has trivial local symmetry group and fixa representative ¯ τ of ¯ τ which coincides, as a typed set, with τ .It is easy to see that Hom( T ¯ τ , h τ i ) consists of only one equivalence class,while Hom( T τ , h ¯ τ i ) consists of two equivalence classes, which we call ϕ and ˜ ϕ . Hom( h τ i , h ¯ τ i ) then consists of the linear span of a “section” Φ such that, restrictingto the representative τ , Φ τ = ϕ + ˜ ϕ , since the action of Z on τ swaps ϕ and ˜ ϕ . A space assignment V for L is a tuple of vector spaces V = ( V t ) t ∈ L . We say aspace assignment V is finite-dimensional if dim( V t ) < ∞ for every t ∈ L . For therest of this subsection we fix an arbitrary (not necessarily finite-dimensional) spaceassignment ( V t ) t ∈ L .For any (finite) typed set T , we write V ⊗ T for the tensor product defined asthe linear span of elementary tensors of the form v = N x ∈ T v x with v x ∈ V t ( x ) , By convention, we always draw the root at the bottom of the tree. egularity structures for vector-valued noises subject to the usual identifications suggested by the notation. Given ψ ∈ Iso( T, ¯ T ) for two typed sets, we can then interpret it as a linear map V ⊗ T → V ⊗ ¯ T by v = O x ∈ T v x ψ · v = O y ∈ ¯ T v ψ − ( y ) . (5.4)In particular, given a symmetric set s , elements a, b ∈ A s , and γ ∈ Γ a,b s , we view γ as a map from V ⊗ T b s to V ⊗ T a s . We then define the vector space V ⊗ s ⊂ Q a ∈ A s V ⊗ T a s by V ⊗ s = n ( v ( a ) ) a ∈ A s : v ( a ) = γ a,b · v ( b ) ∀ a, b ∈ A s , ∀ γ a,b ∈ Γ a,b s o . (5.5)Note that for every a ∈ A s , we have a natural symmetrisation map π s ,a : V ⊗ T a s → V ⊗ s given by ( π s ,a v ) ( b ) = 1 | Γ b,a s | X γ ∈ Γ b,a s γ · v , (5.6)an important property of which is that π s ,a ◦ γ a,b = π s ,b , ∀ a, b ∈ A s , ∀ γ a,b ∈ Γ a,b s . (5.7)Furthermore, these maps are left inverses to the natural inclusions ι s ,a : V ⊗ s → V ⊗ T a s given by ( v ( b ) ) b ∈ A s v ( a ) . Remark 5.14
Suppose that we are given a symmetric set s . For each a ∈ A s , ifwe view s a as a symmetric set in its own right with A s a = { a } , then V ⊗ s a is apartially symmetrised tensor product. In particular V ⊗ s a is again characterised bya universal property, namely it allows one to uniquely factorise multilinear mapson V T a s that are, for every γ ∈ Γ a,a s , γ -invariant in the sense that they are invariantunder a permutation of their arguments like (5.4) with ψ = γ .This construction (where | A s | = 1 ) is already enough to build the vector spacesthat we would want to associate to combinatorial trees as described in Section 5.1.A concrete combinatorial tree, that is a tree with a fixed vertex set and edge setalong with an associated type map, will allow us to construct a symmetric set with | A s | = 1 .We now turn to another feature of our construction, namely that we allow | A s | > . The main motivation is that when we work with combinatorial trees,what we really want is to work with are isomorphism classes of such trees , andso we want our construction to capture that we can allow for many different waysfor the same concrete combinatorial tree to be realised. In particular we will use A s to index a variety of different ways to realise the same combinatorial trees asa concrete set of vertices and edges with type map. The sets Γ a,b s then encode aparticular set of chosen isomorphisms linking different combinatorial trees in thesame isomorphism class. Once they are fixed, the maps (5.6) allow us to movebetween the different vector spaces that correspond to different concrete realisations egularity structures for vector-valued noises of our combinatorial trees. In particular, once s has been fixed, for every a, b ∈ A s one has fixed canonical isomorphisms V ⊗ s a ≃ V ⊗ s b ≃ V ⊗ s (5.8)which can be written explicitly using the maps π s , • of (5.6).In addition to meaningfully resolving the ambiguity between working with iso-morphism classes of objects like trees and concrete instances in those isomorphismclasses, this flexibility is crucial for the formulation and proof of Proposition 5.28. Remark 5.15
Given a space assignment V there is a natural notion of a dual spaceassignment given by V ∗ = ( V ∗ t ) t ∈ L . There is then a canonical inclusion ( V ∗ ) ⊗ s ֒ → ( V ⊗ s ) ∗ . (5.9)Thanks to (5.8) it suffices to prove (5.9) when A s = { a } .Let ι be the canonical inclusion from ( V ∗ ) ⊗ T a s into ( V ⊗ T a s ) ∗ and let r be thecanonical surjection from ( V ⊗ T a s ) ∗ to ( V ⊗ s ) ∗ . The desired inclusion in (5.9) isthen given by the restriction of r ◦ ι to ( V ∗ ) ⊗ s . To the see the claimed injectivity ofthis map, suppose that for some w ∈ ( V ∗ ) ⊗ s one has ι ( w )( v ) = 0 for all v ∈ V ⊗ s .Then, we claim that ι ( w ) = 0 since for arbitrary v ′ ∈ V ⊗ T a s we have ι ( w )( v ′ ) = ι (cid:16) | Γ a,a s | − X γ ∈ Γ a,a s γ · w (cid:17) ( v ′ ) = ι ( w ) (cid:16) | Γ a,a s | − X γ ∈ Γ a,a s γ − · v ′ (cid:17) = 0 where in the first equality we used that w ∈ ( V ∗ ) ⊗ s while in the last equality weused that the sum in the expression before is in V ⊗ s .If the space assignment V is finite-dimensional then our argument above showsthat we have a canonical isomorphism ( V ∗ ) ⊗ s ≃ ( V ⊗ s ) ∗ . (5.10) Example 5.16
Continuing with Example 5.13 and denoting s = h τ i , given vectorspaces V , V , an element in V ⊗ s can be identified with a formal sum over allrepresentations of vectors of the form v x ⊗ v y ⊗ v z + v y ⊗ v x ⊗ v z ,namely, it is partially symmetrised such that for another representation, the twochoices of γ both satisfy the requirement in (5.5) . The projection π s ,a then playsthe role of symmetrisation. We now fix two symmetric sets s and ¯ s . Given Φ ∈ Hom( T a s , ¯ s ) , it naturallydefines a linear map F a Φ : V ⊗ T a s → V ⊗ ¯ s by F a Φ v = π ¯ s , ¯ a (Φ ( ¯ a ) · v ) , (5.11) See for instance Remarks 5.29 and 5.30. egularity structures for vector-valued noises where ¯ a ∈ A ¯ s and Φ ( ¯ a ) denotes any representative of Φ in Iso( T a s , T ¯ a ¯ s ) . Since anyother such choice ¯ b and Φ ( ¯ b ) is related to the previous one by composition to theleft with an element of Γ ¯ a, ¯ b ¯ s , it follows from (5.7) that (5.11) is independent of thesechoices. We extend (5.11) to Vec ( Hom( T a s , ¯ s ) ) by linearity.Since ( ϕ ◦ ψ ) · v = ϕ · ( ψ · v ) by the definition (5.4), we conclude that, for v = ( v ( a ) ) a ∈ A s ∈ V ⊗ s , Φ = ( Φ a ) a ∈ A s ∈ Hom( s , ¯ s ) , as well as γ a,b ∈ Γ a,b s , we havethe identity F b Φ b v ( b ) = F b Φ a ◦ γ a,b v ( b ) = F a Φ a ( γ a,b · v ( b ) ) = F a Φ a v ( a ) ,so that F Φ is well-defined as a linear map from V ⊗ s to V ⊗ ¯ s by F Φ v = F a Φ a ι s ,a v , (5.12)which we have just seen is independent of the choice of a .The following lemma shows that this construction defines a monoidal functor F V mapping s to V ⊗ s and Φ to F Φ between the category SSet of symmetric setsand the category
Vec of vector spaces.
Lemma 5.17
Consider symmetric sets s , ¯ s , ¯¯ s , and morphisms Φ ∈ Hom( s , ¯ s ) and ¯Φ ∈ Hom( ¯ s , ¯¯ s ) . Then F ¯Φ ◦ Φ = F ¯Φ ◦ F Φ .Proof. Since we have by definition F ¯Φ F Φ v = F ¯Φ π ¯ s , ¯ a (Φ ( ¯ a ) a · ι s ,a v ) = π ¯¯ s , ¯¯ a ( ¯Φ ( ¯¯ a ) ¯ b · ι ¯ s , ¯ b π ¯ s , ¯ a (Φ ( ¯ a ) a · ι s ,a v )) ,for any arbitrary choices of a ∈ A s , ¯ a, ¯ b ∈ A ¯ s , ¯¯ a ∈ A ¯¯ s , it suffices to note that ¯Φ ( ¯¯ a ) ¯ b · ι ¯ s , ¯ b π ¯ s , ¯ a w = 1 | Γ ¯ b, ¯ a ¯ s | X γ ∈ Γ ¯ b, ¯ a ¯ s ( ¯Φ ( ¯¯ a ) ¯ b ◦ γ ) · w = ¯Φ ( ¯¯ a ) ¯ a · w , ∀ w ∈ V ⊗ T ¯ a ¯ s ,as an immediate consequence of the definition of Hom( ¯ s , ¯¯ s ) . Remark 5.18
A useful property is the following. Given two space assignments V and W and a collection of linear maps U t : V t → W t , this induces a naturaltransformation F V → F W . Indeed, for any typed set s , it yields a collection oflinear maps U a s : V ⊗ T a s → W ⊗ T a s by U a s O x ∈ T a s v x = O x ∈ T a s U t a s ( x ) v x . This in turn defines a linear map U s : V ⊗ s → W ⊗ s in the natural way. It is thenimmediate that, for any Φ ∈ Hom( s , ¯ s ) , one has the identity U ¯ s ◦ F V ( Φ ) = F W ( Φ ) ◦ U s ,so that this is indeed a natural transformation. egularity structures for vector-valued noises In the more general context of Remark 5.5, this construction proceedssimilarly. The only difference is that now a space assignment V is a functor L → Vec mapping objects t to spaces V t and morphisms ϕ ∈ Hom L ( t , ¯ t ) to linearmaps V ϕ ∈ L ( V t , V ¯ t ) . In this case, an element ϕ ∈ Iso( T, ¯ T ) naturally yields alinear map V ⊗ T → V ⊗ ¯ T by v = O x ∈ T v x ϕ · v = O y ∈ ¯ T V ~ϕ ( ϕ − ( y )) v ϕ − ( y ) . The remainder of the construction is then essentially the same.
Remark 5.20
One may want to restrict oneself to a smaller category than
Vec byenforcing additional “nice” properties on the spaces V t . For example, it will beconvenient below to replace it by some category of topological vector spaces. It will be convenient to consider the larger category
TStruc of typed structures . Definition 5.21
We define
TStruc to be the category obtained by freely adjoin-ing countable products to
SSet . We write
TStruc L for TStruc when we want toemphasize the dependence of this category on the underlying label set L . Remark 5.22
An object S in the category TStruc can be viewed as a countable(possibly finite) index set A and, for every α ∈ A , a symmetric set s α ∈ Ob(SSet) .This typed structure is then equal to Q α ∈A s α , where Q denotes the categoricalproduct. (When A is finite it coincides with the coproduct and we will then alsowrite L α ∈A s α and call it the “direct sum” in the sequel.) Morphisms between S and ¯ S can be viewed as “infinite matrices” M ¯ α,α with α ∈ A , ¯ α ∈ ¯ A , M ¯ α,α ∈ Hom( s α , s ¯ α ) and the property that, for every ¯ α ∈ ¯ A , one has M ¯ α,α = 0 for all butfinitely many values of α . Composition of morphisms is performed in the naturalway, analogous to matrix multiplication.We remark that the index set A here has nothing to do with the index set A s inDefinition 5.2.Note that TStruc is still symmetric monoidal with the tensor product behav-ing distributively over the direct sum if we enforce ( Q α ∈A s α ) ⊗ ( Q β ∈B s β ) = Q ( α,β ) ∈A×B ( s α ⊗ s β ) , with s α ⊗ s β as in Remark 5.10, and define the tensorproduct of morphisms in the natural way. Remark 5.23
The choice of adjoining countable products (rather than coproducts)is that we will use this construction in Section 5.8 to describe the general solutionto the algebraic fixed point problem associated to (5.1) as an infinite formal series. egularity structures for vector-valued noises If the space assignments V t are finite-dimensional, the functor F V then naturallyextends to an additive monoidal functor from TStruc to the category of topologicalvector spaces (see for example [ST12, Sec. 4.5]). Note that in particular one has F V ( S ) = Q α ∈A F V ( s α ) . In Section 5.2 we showed how, given a set of labels L and space assignment ( V t ) t ∈ L ,we can “extend” this space assignment so that we get an appropriately symmetrisedvector space V ⊗ s for any symmetric set s (or, more generally, for any typed structure)typed by L . In this subsection we will investigate how this construction behavesunder a direct sum decomposition for the space assignment ( V t ) t ∈ L that is encodedvia a corresponding “decomposition” on the set L . Definition 5.24
Let P ( A ) denote the powerset of a set A . Given two distinct finitesets of labels L and ¯ L as well as a map p : L → P ( ¯ L ) \{6 } , such that { p ( t ) : t ∈ L } is a partition of ¯ L , we call ¯ L a type decomposition of L under p . If we are alsogiven space assignments ( V t ) t ∈ L for L and ( ¯ V l ) l ∈ ¯ L for ¯ L with the property that V t = M l ∈ p ( t ) ¯ V l for every t ∈ L , (5.13)then we say that ( ¯ V l ) l ∈ ¯ L is a decomposition of ( V t ) t ∈ L . For l ∈ p ( t ) , we write P l : V t → ¯ V l for the projection induced by (5.13).For the remainder of this subsection we fix a set of labels L , a space assignment ( V t ) t ∈ L , along with a type decomposition ¯ L of L under p and a space assignment ( ¯ V l ) l ∈ ¯ L that is a decomposition of ( V t ) t ∈ L . To shorten notations, for functions t : B → L and l : B → ¯ L with any set B , we write l t as a shorthand for therelation l ( p ) ∈ p ( t ( p )) for every p ∈ B . Given any symmetric set s with label set L and any a ∈ A s , we write ˆ L a s = { l : T a s → ¯ L : l t a s } , and we consider on ˆ L s = S a ∈ A s ˆ L a s the equivalence relation ∼ given by ˆ L a s ∋ l ∼ ¯ l ∈ ˆ L b s ⇔ ∃ γ b,a ∈ Γ b,a s : l = ¯ l ◦ γ b,a . (5.14)We denote by L s def = ˆ L s / ∼ the set of such equivalence classes, which we note isfinite. Example 5.25
In this example we describe the vector space associated to . Forour space assignment we start with a labelling set L = { ⋆ } , where ⋆ representsthe noise, and the noise takes values in a vector space V ⋆ . If we wanted to expandour noise into two components we can encode this via a direct sum decomposition V ⋆ = V ( ⋆, ) ⊕ V ( ⋆, ) , where we introduce a new set of labels ¯ L = L × { , } andwe define p ( ⋆ ) = { ( ⋆, ) , ( ⋆, ) } .Recall our convention described in Example 5.12: represents an isomor-phism class of trees and any concrete tree τ in that class is realised by a vertex egularity structures for vector-valued noises set which is a subset of 3 elements of N and in which 2 of those elements are theleaves labelled by ⋆ . Let s be the symmetric set associated to and τ ∈ A s be aconcrete tree with T τ = { x, y } , t τ ( x ) = t τ ( y ) = ⋆ . Then the local symmetry groupis isomorphic to Z .The set ˆ L τ s consists of 4 elements which we denote by , , , and . (5.15) In the symbols above, the left leaf corresponds to x and the right one to y . Wethus obtain four labellings on T τ by ¯ L where ( ⋆, ) is associated to and ( ⋆, ) isassociated to .However, if we were to interpret the symbols of (5.15) as isomorphism classesof trees labelled by ¯ L then and are the same isomorphism class and thisis reflected by the fact that | L s | = 3 . The isomorphism class of is associatedto a vector space isomorphic to V ⋆ ⊗ s V ⋆ . Our construction will decompose (seeProposition 5.28) the vector space for into a direct sum of three vector spacescorresponding to the isomorphism classes , , and which are, respectively,isomorphic to V ( ⋆, ) ⊗ s V ( ⋆, ) , V ( ⋆, ) ⊗ s V ( ⋆, ) , and V ( ⋆, ) ⊗ V ( ⋆, ) . Given an equivalence class Y ∈ L s and a ∈ A s , we define Y a = Y ∩ ˆ L a s (whichwe note is non-empty due to the connectedness of Γ s ). We then define a symmetricset s Y by A s Y = { ( a, l ) : a ∈ A s , l ∈ Y a } , T ( a, l ) s Y = T a s , t ( a, l ) s Y = l , Γ ( a, l ) , ( b, ¯ l ) s Y = { γ ∈ Γ a,b s : ¯ l = l ◦ γ } . Remark 5.26
The definition (5.14) of our equivalence relation guarantees that Γ s Y is connected. The definition of Γ s Y furthermore yields a morphism of groupoids Γ s Y → Γ s which is easily seen to be surjective.With these notations at hand, we can define a functor p ∗ from SSet L to TStruc ¯ L asfollows. Given any s ∈ Ob(SSet L ) , we define p ∗ s = M Y ∈ L s s Y ∈ Ob ( TStruc ¯ L ) . (5.16)To describe how p ∗ acts on morphisms, let us fix two symmetric sets s , ¯ s ∈ Ob(SSet L ) , a choice of Y ∈ L s , ( a, l ) ∈ A s Y , as well as an element ϕ ∈ Hom( T a s , ¯ s ) .We then let ϕ · l ⊂ ˆ L ¯ s be given by ϕ · l = l ◦ ϕ − def = { ¯ l : ∃ ψ ∈ ϕ with ¯ l = l ◦ ψ − } ,where we recall that ϕ ⊂ S ¯ a ∈ A ¯ s Iso( T a s , T ¯ a ¯ s ) is a Γ ¯ s -equivalence class of bijections.It follows from the definitions of the equivalence relation on ˆ L ¯ s and of Hom( T a s , ¯ s ) egularity structures for vector-valued noises that one actually has ϕ · l ∈ L ¯ s . We then define p ∗ ( a, l ) ϕ ∈ M ¯ Y ∈ L ¯ s Vec ( Hom( T ( a, l ) s Y , ¯ s ¯ Y ) ) ,by simply setting p ∗ ( a, l ) ϕ = ϕ ∈ Hom( T ( a, l ) s Y , ¯ s ϕ · l ) ⊂ M ¯ Y ∈ L ¯ s Vec ( Hom( T ( a, l ) s Y , ¯ s ¯ Y ) ) , (5.17)which makes sense since T ( a, l ) s Y = T a s , T ( ¯ a, ¯ l ) ¯ s ¯ Y = T ¯ a ¯ s , and since Γ ¯ s ϕ · l → Γ ¯ s issurjective.We take a moment to record an important property of this construction. Givenany ( a, l ) , ( b, ˆ l ) ∈ A s Y , γ ∈ Γ ( a, l ) , ( b, ˆ l ) s Y , and ϕ ∈ Hom( T a s , ¯ s ) , it follows from ourdefinitions that ( p ∗ ( a, l ) ϕ ) ◦ γ = p ∗ ( b, ˆ l ) ( ϕ ◦ γ ) (5.18)where on the right-hand side of (5.18) we are viewing γ as an element of Γ a,b s ,which indeed maps Hom( T a s , ¯ s ) into Hom( T b s , ¯ s ) by right composition.Extending (5.17) by linearity, we obtain a map p ∗ ( a, l ) : Vec ( Hom( T a s , ¯ s ) ) → M ¯ Y ∈ L ¯ s Vec ( Hom( T ( a, l ) s Y , ¯ s ¯ Y ) ) . We then use this to construct a map p ∗ Y : Hom( s , ¯ s ) → Hom( s Y , p ∗ ¯ s ) as follows.For any Φ = ( Φ a ) a ∈ A s ∈ Hom( s , ¯ s ) , we set ( p ∗ Y Φ ) ( a, l ) = p ∗ ( a, l ) Φ a , ∀ ( a, l ) ∈ A s Y . To show that this indeed belongs to
Hom( s Y , p ∗ ¯ s ) , note that, for any ( a, l ) , ( b, ˆ l ) ∈ A s Y and γ ∈ Γ ( a, l ) , ( b, ˆ l ) s Y ⊂ Γ a,b s , we have ( p ∗ Y Φ ) ( a, l ) ◦ γ = ( p ∗ ( a, l ) Φ a ) ◦ γ = ( p ∗ ( b, ˆ l ) ( Φ a ◦ γ ) ) = ( p ∗ ( b, ˆ l ) Φ b ) = ( p ∗ Y Φ ) ( b, ˆ l ) . In the second equality we used the property (5.18) and in the third equality we usedthat Φ a ◦ γ = Φ b which follows from our assumption that Φ ∈ Hom( s , ¯ s ) – recallthat here we are viewing γ as an element of Γ a,b s .Finally, we then obtain the desired map p ∗ : Hom( s , ¯ s ) → Hom( p ∗ s , p ∗ ¯ s ) bysetting, for Φ ∈ Hom( s , ¯ s ) , p ∗ Φ = M Y ∈ L s p ∗ Y Φ . The fact that p ∗ is a functor (i.e. preserves composition of morphisms) is analmost immediate consequence of (5.17). Indeed, given ϕ ∈ Hom( T a s , ¯ s ) and ¯Φ ∈ Hom( ¯ s , ¯¯ s ) , it follows immediately from (5.17) that p ∗ ϕ · l ¯Φ ◦ p ∗ ( a, l ) ϕ = p ∗ ( a, l ) ( ¯Φ ◦ ϕ ) , egularity structures for vector-valued noises where we view ¯Φ ◦ ϕ as an element of Vec ( Hom( T a s , ¯¯ s ) ) . It then suffices to notethat p ∗ ¯ Y ¯Φ ◦ p ∗ ( a, l ) ϕ = 0 for ¯ Y = ϕ · l , which then implies that p ∗ ¯Φ ◦ p ∗ ( a, l ) ϕ = p ∗ ( a, l ) ( ¯Φ ◦ ϕ ) ,and the claim follows. Note also that p ∗ is monoidal in the sense that p ∗ ( s ⊗ ¯ s ) = p ∗ ( s ) ⊗ p ∗ ( ¯ s ) and similarly for morphisms, modulo natural transformations. Remark 5.27
One property that can be verified in a rather straightforward way isthat if we define ( ¯ p ◦ p )( t ) = S l ∈ p ( t ) ¯ p ( l ) , then ( ¯ p ◦ p ) ∗ = ¯ p ∗ ◦ p ∗ ,again modulo natural transformations. This is because triples ( a, l , ¯ l ) with ¯ l l t a are in natural bijection with pairs ( a, ¯ l ) . Note that this identity crucially uses thatthe sets p ( t ) are all disjoint.Our main interest in the functor p ∗ is that it will perform the corresponding directsum decompositions at the level of partially symmetric tensor products of the spaces V t . This claim is formulated as the following proposition. Proposition 5.28
One has F ¯ V ◦ p ∗ = F V , modulo natural transformation.Proof. Fix s ∈ SSet L . Given any a ∈ A s and elementary tensor v ( a ) ∈ V ⊗ T a s ofthe form v ( a ) = N x ∈ T a s v ( a ) x , we first note that we have the identity v ( a ) = O x ∈ T a s X l ∈ p ( t s ( x )) P l v ( a ) x = X l t s O x ∈ T a s P l ( x ) v ( a ) x , (5.19)where P l is defined below (5.13). This suggests the following definition for a map ι s : Y a ∈ A s V ⊗ T a s → Y ( a, l ) ∈ ˆ L s ¯ V ⊗ T ( a, l ) s [ a, l ] ,where [ a, l ] ∈ L s is the equivalence class that ( a, l ) belongs to. Given v = ( v ( a ) ) a ∈ A s with v ( a ) = N x ∈ T a s v ( a ) x , we set ( ι s v ) ( a, l ) def = O x ∈ T a s P l ( x ) v ( a ) x ,which is clearly invertible with inverse given by ( ι − s w ) a = P l t a N x ∈ T a s w ( a, l ) x .Note now that F V ( s ) ⊂ Y a ∈ A s V ⊗ T a s , F ¯ V ( p ∗ s ) = M Y ∈ L s F ¯ V ( s Y ) ⊂ M Y ∈ L s Y ( a, l ) ∈ A s Y ¯ V ⊗ T ( a, l ) s Y ≃ Y ( a, l ) ∈ ˆ L s ¯ V ⊗ T ( a, l ) s [ a, l ] , egularity structures for vector-valued noises where we used that L s is finite in the final line. Furthermore, both ι s and ι − s preserve these subspaces, and we can thus view ι s as an isomorphism of vectorspaces between F V ( s ) and F ¯ V ( p ∗ s ) . The fact that, for Φ ∈ Hom( s , ¯ s ) , one has ι ¯ s ◦ F V ( Φ ) = F ¯ V ( p ∗ Φ ) ◦ ι s ,is then straightforward to verify. Most of the symmetric sets entering our constructions will be generated from finite labelled rooted trees (sometimes just called “trees” for simplicity) and theirassociated automorphisms. A finite labelled rooted tree τ = ( T, ̺, t , n ) consists ofa tree T = ( V, E ) with finite vertex set V , edge set E ⊂ V × V and root ̺ ∈ V ,endowed with a type t : E → L and label n : V ∪ E → N d +1 . We also write e : E → L × N d +1 for the map e = ( t , n ↾ E ) . Note that the “smallest” possible tree,usually denoted by , is given by V = { ̺ } , E = and n ( ̺ ) = 0 ; we denote by X k with k ∈ N d +1 the same tree but with n ( ̺ ) = k . For convenience, we consideredges as directed towards the root in the sense that we always have e = ( e − , e + ) with e + nearer to the root. Note that one can naturally extend the map t to V \ { ̺ } by setting t ( v ) = t ( e ) for the unique edge e such that e − = v .An isomorphism between two labelled rooted trees is a bijection between theiredge and vertex sets that preserves their connective structure, their roots, and theirlabels t and n . We then denote by T the set of isomorphism classes of rootedlabelled trees with vertex sets that are subsets of N . Given τ ∈ T , we assign to it a symmetric set s = h τ i . In particular, wefix A s = τ and, for every τ ∈ τ , we set T τ s = E τ (the set of edges of τ ), t τ s the type map of τ , and, for τ , τ ∈ τ , we let Γ τ ,τ s be the set of all elementsof Iso( T τ s , T τ s ) obtained from taking a tree isomorphism from τ to τ and thenrestricting this map to the set of edges E τ . We also define the object h T i in TStruc given by h T i = Q τ ∈ T h τ i .Given an arbitrary labelled rooted tree τ , we also write h τ i for the symmetricset with A h τ i a singleton, T h τ i and t h τ i as above, and Γ the set of all automorphismsof τ . The following remark is crucial for our subsequent use of notations. Remark 5.29
By definition, given any labelled rooted tree τ , there exists exactlyone τ ∈ T such that its elements are tree isomorphic to τ and exactly one elementof Hom( T h τ i , h τ i ) whose representatives are tree isomorphisms, so we are in thesetting of Remark 5.11. As a consequence, we can, for all intents and purposes,identify h τ i with h τ i . As an example, in Section 5.4.1 this observation allows us todefine various morphisms on h T i and h T i ⊗ h T i by defining operations at the levelof trees τ ∈ τ ∈ T with fully specified vertex and edge sets rather than workingwith the isomorphism class τ . The choice of N here is of course irrelevant; any set of infinite cardinality would do. The onlyreason for this restriction is to make sure that elements of T are sets. egularity structures for vector-valued noises In the example in Section 5.1, the two labelled rooted trees τ = and ¯ τ = (both with trivial labels n = 0 say) are isomorphic. The canonicalisomorphism Φ ∈ Hom( h τ i , h ¯ τ i ) is then simply the map matching the same types.The map F V ( Φ ) : V ⊗h τ i → V ⊗h ¯ τ i is then a canonical isomorphism – this is wherethe middle identity in the motivation (5.2) is encoded.A labelled rooted forest f = ( F, P , t , n ) is defined as consisting of a finite forest F = ( V, E ) , where again V is the set of vertices, E the set of edges, each connectedcomponent T of F has a unique distinguished root ̺ with P ⊂ V the set of all theseroots, and t and n are both as before. Note that we allow for the empty forest, thatis the case where V = E = , and that any finite labelled rooted tree ( T, ̺, t , n ) isalso a forest (where P = { ̺ } ).Two labelled rooted forests are considered isomorphic if there is a bijectionbetween their edge and vertex sets that preserves their connective structure, theirroots, and their labels t and n – note that we allow automorphisms of labelled rootedforests to swap connected components of the forest. Given a labelled rooted forest f , we then write h f i for the corresponding symmetric set constructed similarly toabove, now with tree automorphisms replaced by forest automorphisms.We denote by F the set of isomorphism classes of rooted labelled forests withvertex sets in N , which can naturally be viewed as the unital commutative monoidgenerated by T with unit given by the empty forest. In the same way as above, weassign to an element f ∈ F a symmetric set h f i and we write h F i = Q f ∈ F h f i ∈ Ob(TStruc) .Before continuing our discussion we take a moment to describe where we aregoing. In many previous works on regularity structures, in particular in [BHZ19],the vector space underlying a regularity structure is given by
Vec( T ( R )) for a subset T ( R ) ⊂ T determined by some rule R . The construction and action of the structureand renormalisation groups was then described by using combinatorial operationson elements of T and F .Here our point of view is different. Our concrete regularity structure will beobtained by applying the functor F V to h T ( R ) i , an object in TStruc . We will referto h T ( R ) i as an “abstract” regularity structure. In particular, trees τ ∈ T ( R ) willnot be interpreted as basis vectors for our regularity structure anymore, but insteadserve as an indexing set for subspaces canonically isomorphic to F V ( h τ i ) . In thecase when V t ≃ R for all t ∈ L , this is of course equivalent, but in general it isnot. Operations like integration, tree products, forest products, and co-productson the regularity structures defined in [BHZ19] were previously given in termsof operations on T and / or F . In order to push these operations to our concreteregularity structure, we will in the next section describe how to interpret them asmorphisms between the corresponding typed structures, which then allows us topush them through to “concrete” regularity structures using F V . Recall that a forest is a graph without cycles, so that every connected component is a tree. egularity structures for vector-valued noises
Although the definition of T L depends on the choice of L , thisdefinition is compatible with p ∗ in the following sense. Given τ ∈ T L , if L h τ i isdefined as in the definition immediately below (5.14), then L h τ i can be identifiedwith a subset of T ¯ L . In particular, we overload notation and define a map p : T L →P ( T ¯ L ) \ {6 } by setting p ( τ ) = L h τ i so p ∗ h τ i ≃ L ¯ τ ∈ p ( τ ) h ¯ τ i . It is also easy tosee that { p ( τ ) : τ ∈ T L } is a partition of T ¯ L so that p ∗ h T L i ≃ h T ¯ L i . Analogous statements hold for the sets of forests F L and F ¯ L . We start by recalling the tree product. Given two rooted labelled trees τ = ( T, ̺, t , n ) and ¯ τ = ( ¯ T , ¯ ̺, ¯ t , ¯ n ) the tree product of τ and ¯ τ , which we denote τ ¯ τ , is a rootedlabelled tree defined as follows. Writing τ ¯ τ = ( ˆ T , ˆ ̺, ˆ t , ˆ n ) , one sets ˆ T def = ( T ⊔ ¯ T ) / { ̺, ¯ ̺ } , namely ˆ T is the rooted tree obtained by taking the rooted trees T and ¯ T and identifying the roots ̺ and ¯ ̺ into a new root ˆ ̺ . Writing T = ( V, E ) , ¯ T = ( ¯ V , ¯ E ) ,and ˆ T = ( ˆ V , ˆ E ) , we have a canonical identification of ˆ E with E ⊔ ¯ E and ˆ V \ { ˆ ̺ } with ( V ⊔ ¯ V ) \ { ̺, ¯ ̺ } . With these identifications in mind, ˆ t is obtained from theconcatenation of t and ¯ t . We also set ˆ n ( a ) def = n ( a ) if a ∈ E ⊔ ( V \ { ̺ } ) , ¯ n ( a ) if a ∈ ¯ E ⊔ ( ¯ V \ { ¯ ̺ } ) , n ( ̺ ) + ¯ n ( ¯ ̺ ) if a = ˆ ̺ . We remark that the tree product is well-defined and commutative at the level ofisomorphism classes.In order to push this tree product through our functor, we want to encode it asa morphism
M ∈
Hom( h T i ⊗ h T i , h T i ) . It is of course sufficient for this to defineelements M ∈
Hom( h τ i ⊗ h ¯ τ i , h τ ¯ τ i ) for any τ , ¯ τ ∈ T , which in turn is given by h τ i ⊗ h ¯ τ i ≃ h τ i ⊗ h ¯ τ i → h τ ¯ τ i ≃ h τ ¯ τ i , (5.20)where the two canonical isomorphisms are the ones given by Remark 5.11 and themorphism in Hom ( h τ i ⊗ h ¯ τ i , h τ ¯ τ i ) is obtained as follows. Note that the same typedset ( ˆ E, ˆ t ) underlies both the symmetric sets h τ i ⊗ h ¯ τ i and h τ ¯ τ i and that Γ h τ i⊗h ¯ τ i is a subgroup (possibly proper) of Γ h τ ¯ τ i . Therefore, the only natural element of Hom( h τ i ⊗ h ¯ τ i , h τ ¯ τ i ) is the equivalence class of the identity in Hom( E ⊔ ¯ E, h τ ¯ τ i ) (in the notation of Remark 5.8). It is straightforward to verify that M constructedin this way is independent of the choices τ ∈ τ and ¯ τ ∈ ¯ τ .The “neutral element” η ∈ Hom( I, h T i ) for M , where I denotes the unit objectin TStruc (corresponding to the empty symmetric set), is given by the canonicalisomorphism I → h i with denoting the tree with a unique vertex and n = 0 as before, composed with the canonical inclusion h i → h T i . One does indeed egularity structures for vector-valued noises have M ◦ ( η ⊗ id) = M ◦ (id ⊗ η ) = id , with equalities holding modulo theidentifications h T i ≃ h T i ⊗ I ≃ I ⊗ h T i . Associativity holds in a similar way,namely M ◦ (id ⊗ M ) = M ◦ ( M ⊗ id) as elements of
Hom( h T i ⊗ h T i ⊗ h T i , h T i ) . Remark 5.32
Another important remark is that the construction of the product M respects the functors p ∗ in the same way as the construction of h T i does.As mentioned above, F is viewed as the free unital commutative monoid generatedby T with unit given by the empty forest (which we denote by ). We can interpretthis product in the following way. Given two rooted labelled forests f = ( F, P , t , n ) and ¯ f = ( ¯ F , ¯ P , ¯ t , ¯ n ) we define the forest product f · ¯ f = ( ˆ F , ˆ P , ˆ t , ˆ n ) by ˆ F = F ⊔ ¯ F , ˆ P = P ⊔ ¯ P , ˆ t = t ⊔ ¯ t , and ˆ n = n ⊔ ¯ n . Again, it is easy to see that this productis well-defined and commutative at the level of isomorphism classes. As before,writing ˆ F = ( ˆ V , ˆ E ) and noting that the same typed set ( ˆ E, ˆ t ) = ( E ⊔ ¯ E, t ⊔ ¯ t ) underlies both symmetric sets h f i ⊗ h ¯ f i and h f · ¯ f i and that the symmetry groupof the former is a subgroup of that of the latter, there is a natural morphism Hom( h f i ⊗ h ¯ f i , h f · ¯ f i ) given by the equivalence class of the identity. As before,this yields a product morphism in Hom( h F i ⊗ h F i , h F i ) , this time with the canonicalisomorphism between I and h6 i (with the empty forest) playing the role of theneutral element.We now turn to integration. Given any l ∈ L and τ = ( T, ̺, t , n ) ∈ T wedefine a new rooted labelled tree I ( l , ) ( τ ) = ( ¯ T , ¯ ̺, ¯ t , ¯ n ) ∈ T as follows. The tree ¯ T = ( ¯ V , ¯ E ) is obtained from T = ( V, E ) by setting ¯ V def = V ⊔ { ¯ ̺ } and ¯ E = E ⊔ { ¯ e } where ¯ e = ( ̺, ¯ ̺ ) , that is one adds a new root vertex to the tree T and connects itto the old root with an edge. We define ¯ t to be the extension of t to ¯ E obtained bysetting t ( ¯ e ) = l and ¯ n to be the extension of n obtained by setting ¯ n ( ¯ e ) = ¯ n ( ¯ ̺ ) = 0 .We encode this into a morphism I ( l , ) ∈ Hom( h T i ⊗ h l i , h T i ) where h l i denotesthe symmetric set with a single element • of type l . For this, it suffices to exhibitnatural morphisms Hom ( h τ i ⊗ h l i , h I ( l , ) ( τ ) i ) , (5.21)which are given by the equivalence class of ι : E ⊔ {•} → ¯ E in Hom( E ⊔{•} , h I ( l , ) ( τ ) i ) , where ι is the identity on E and ι ( • ) = ¯ e . It is immediatethat this respects the automorphisms of τ and therefore defines indeed an elementof Hom ( h τ i ⊗ h l i , h I ( l , ) ( τ ) i ) . The construction above also gives us correspondingmorphisms I ( l ,p ) ∈ Hom( h T i ⊗ h l i , h T i ) , for any p ∈ N d +1 , if we exploit thecanonical isomorphism h I ( l , ) ( τ ) i ≃ h I ( l ,p ) ( τ ) i where I ( l ,p ) ( τ ) is constructed justas I ( l , ) ( τ ) , the only difference being that one sets ¯ n ( ¯ e ) = p . In order to build a regularity structure, we will also need analogues of the maps ∆ + and ∆ − as defined in [BHZ19]. The following construction will be very useful:given τ ∈ τ ∈ T and f ∈ f ∈ F , we write f ֒ → τ for the specification ofan injective map ι : T f → T τ which preserves connectivity, orientation, and type egularity structures for vector-valued noises (but roots of f may be mapped to arbitrary vertices of τ ). We also impose that n f ( e ) = n τ ( ιe ) for every edge e ∈ E f and that polynomial vertex labels areincreased by ι in the sense that n f ( x ) ≤ n τ ( ιx ) for all x ∈ V f . Given f ֒ → τ , wealso write ∂E f ⊂ E τ \ ι ( E f ) for the set of edges e “incident to f ” in the sense that e + ∈ ι ( V f ) . We consider inclusions ι : f ֒ → τ and ¯ ι : ¯ f ֒ → τ to be “the same” ifthere exists a forest isomorphism ϕ : f → ¯ f such that ι = ¯ ι ◦ ϕ . (We do howeverconsider them as distinct if they differ by a tree isomorphism of the target τ !)Given a label e : ∂E f → N d +1 , we write π e : V f → N d +1 for the map givenby π e ( x ) = P e + = ιx e ( e ) and we write f e for the forest f , but with n f replaced by n f + π e . We then write τ /f e ∈ T for the tree constructed as follows. Its vertex setis given by V τ / ∼ f , where ∼ f is the equivalence relation given by x ∼ f y if andonly if x, y ∈ ι ( V f ) and ι − x and ι − y belong to the same connected component of f . The edge set of τ /f e is given by E τ \ ι ( E f ) , and types and the root are inheritedfrom τ . Its edge label is given by e n τ ( e ) + e ( e ) . Noting that vertices of τ /f e are subsets of V τ , its vertex label is given by x P y ∈ x ( n τ ( y ) − n f ( ι − y ) ) withthe convention that n f is extended additively to subsets. (This is positive by ourdefinition of “inclusion”.)This construction then naturally defines an ‘extraction / contraction’ operation ( f ֒ → τ ) e ∈ Hom( E τ , h f e i ⊗ h τ /f e i ) similarly to above. (Using ι , the edge set of τ is canonically identified with the disjoint union of the edge set of f e with that of τ /f e .) Note that this is well-defined in the sense that two identical (in the sensespecified above) inclusions yield identical (in the sense of canonically isomorphic)elements of Hom( E τ , h f e i ⊗ h τ /f e i ) . We also define ¯ f /f e and ( f ֒ → ¯ f ) e for aforest ¯ f in the analogous way.We also define a “cutting” operation in a very similar way. Given two trees τ and ¯ τ , we write ¯ τ r ֒ → τ if ¯ τ ֒ → τ (viewing ¯ τ as a forest with a single tree) and theinjection ι furthermore maps the root of ¯ τ onto that of τ . With this definition athand, we define “extraction” and “cutting” operators ∆ ex [ τ ] ∈ Hom( E τ , h F i ⊗ h T i ) , ∆ ex [ τ ] = X f֒ → τ X e e ! (cid:18) τf (cid:19) ( f ֒ → τ ) e , ∆ cut [ τ ] ∈ Hom( E τ , h T i ⊗ h T i ) , ∆ cut [ τ ] = X ¯ τ r ֒ → τ X e e ! (cid:18) τ ¯ τ (cid:19) ( ¯ τ r ֒ → τ ) e . Here, the inner sum runs over e : ∂E f → N d ( e : ∂E ¯ τ → N d in the second case)and the binomial coefficient (cid:0) τf (cid:1) is defined as (cid:18) τf (cid:19) = Y x ∈ V f (cid:18) n τ ( ιx ) n f ( x ) (cid:19) . We also view
Hom( E τ , h f e i ⊗ h τ /f e i ) as a subset of Hom( E τ , h F i ⊗ h T i ) via thecanonical maps h τ i ≃ h τ i ֒ → h T i and similarly for h F i . Lemma 5.33
One has ∆ ex [ τ ] ∈ Hom( h τ i , h F i⊗h T i ) as well as ∆ cut [ τ ] ∈ Hom( h τ i , h T i⊗h F i ) . egularity structures for vector-valued noises Proof.
Given any isomorphism ϕ of τ , it suffices to note that, in Hom( h τ i , h F i ⊗h T i ) , we have the identity ( f ֒ → τ ) e ◦ ϕ = ( f ϕ ֒ → τ ) e ϕ ,where, if f ֒ → τ is represented by ι , then f ϕ ֒ → τ is represented by ϕ − ◦ ι and e ϕ = e ◦ ϕ . It follows that ∆ ex [ τ ] ◦ ϕ = ∆ ex [ τ ] as required. The argument for ∆ cut is virtually identical.It also follows from our construction that, given τ ∈ T , ∆ ex [ τ ] and ∆ cut [ τ ] areindependent of τ ∈ τ , modulo canonical isomorphism as in Remark 5.11 (see also(5.20) above), so that we can define ∆ ex [ τ ] ∈ Hom( h τ i , h F i ⊗ h T i ) and similarlyfor ∆ cut . We now show how regularity structures generated by rules as in [BHZ19] can berecast in this framework. This then allows us to easily formalise constructions ofthe type “attach a copy of V to every noise / kernel” as was done in a somewhat adhoc fashion in [GH19, Sec. 3.1].We will restrict ourselves to the setting of reduced abstract regularity structures(as in the language of [BHZ19, Section 6.4]). The extended label (as in [BHZ19,Section 6.4]) will not play an explicit role here but appears behind the sceneswhen we use the black box of [BHZ19] to build a corresponding scalar reducedregularity structure, which is then identified, via the natural transformation ofProposition 5.28, with the concrete regularity structure obtained by applying F V toour abstract regularity structure.We start by fixing a degree map deg : L → R (where L was our previouslyfixed set of labels), a “space” dimension d ∈ N , and a scaling s ∈ [ , ∞ ) d +1 .Multiindices k ∈ N d +1 are given a scaled degree | k | s = P di =0 k i s i . (We use theconvention that the -component denotes the time direction.)We then define, as in [BHZ19, Eq. (5.5)], the sets E of edge labels and N ofnode types by E = L × N d +1 , N = ˆ P ( E ) ,where ˆ P ( A ) denotes the set of all multisets with elements from A . With thisnotation, we fix a “rule” R : L → P ( N ) \ {6 } . We will only consider rules thatare subcritical and complete in the sense of [BHZ19, Def. 5.22].Given τ ∈ τ ∈ T with underlying tree T = ( V, E ) , every vertex v ∈ V isnaturally associated with a node type N ( v ) def = ( o ( e ) : e + = v ) ∈ N , where we set o ( e ) = ( t ( e ) , n ( e )) ∈ E . The degree of τ is given by deg τ = X v ∈ V | n ( v ) | s + X e ∈ E ( deg t ( e ) − | n ( e ) | s ) . We say that τ strongly conforms to R if egularity structures for vector-valued noises (i) for every v ∈ V \ { ̺ } , one has N ( v ) ∈ R ( t ( v )) , and(ii) there exists t ∈ L such that N ( ̺ ) ∈ R ( t ) .We say that τ is planted if ♯ N ( ̺ ) = 1 and n ( ̺ ) = 0 and unplanted otherwise.The map deg , the above properties, and the label n ( ̺ ) ∈ N d +1 depend only onthe isomorphism class τ ∋ τ , and we shall use the same terminology for τ . Wewrite T ( R ) ⊂ T for the set of τ that strongly conform to R . We further write T ⋆ ( R ) ⊂ T for the set of planted trees satisfying condition (i).We now introduce the algebras of trees / forests that are used for negative andpositive renormalisation. We write F ( R ) ⊂ F for the unital monoid generated (forthe forest product) by T ( R ) , F − ( R ) ⊂ F ( R ) for the unital monoid generated by T − ( R ) def = { τ ∈ T ( R ) : deg τ < , n ( ̺ ) = 0 , τ unplanted } , (5.22)and T + ( R ) ⊂ T for the unital monoid generated (for the tree product) by { X k : k ∈ N d } ∪ { τ ∈ T ⋆ ( R ) : deg τ > } . We are now ready to construct our abstract regularity structure in the category
TStruc that was defined in Definition 5.21. We define T , T + , T − , F , F − ∈ Ob(TStruc) by T def = h T ( R ) i , T + def = h T + ( R ) i , T − def = h T − ( R ) i , F def = h F ( R ) i , F − def = h F − ( R ) i . As mentioned earlier, we think of T as an “abstract” regularity structure with“characters on T + ” forming its structure group and “characters on F − ” forming itsrenormalisation group. Write π + ∈ Hom( T , T + ) , π − ∈ Hom( F , F − ) ,for the natural projections. These allow us to define Hom( T , T ⊗ T + ) ∋ ∆ + def = X τ ∈ T ( R ) (id ⊗ π + ) ∆ cut [ τ ] , Hom( T , F − ⊗ T ) ∋ ∆ − def = X τ ∈ T ( R ) ( π − ⊗ id) ∆ ex [ τ ] . Remark 5.34
Here and below, it is not difficult to see that these expressions doindeed define morphisms of
TStruc . Regarding ∆ + for example, it suffices tonote that, given any τ ∈ T ( R ) and τ + ∈ T + ( R ) , there exist only finitely manypairs of trees τ ( ) r ֒ → τ ( ) in T ( R ) and edge labels e in Section 5.4.2, such that τ + = τ ( ) / τ ( ) e and τ = τ ( ) e .In an analogous way, we also define ∆ + s ∈ Hom( T + , T + ⊗ T + ) , ∆ − s ∈ Hom( F − , F − ⊗ F − ) , egularity structures for vector-valued noises by ∆ + s def = X τ ∈ T + ( R ) ( π + ⊗ π + ) ∆ cut [ τ ] , ∆ − s def = X f ∈ F − ( R ) ( π − ⊗ π − ) ∆ ex [ f ] . As in [BHZ19], one has the identities (∆ − s ⊗ id ) ◦ ∆ − = ( id ⊗ ∆ − ) ◦ ∆ − , (∆ − s ⊗ id ) ◦ ∆ − = ( id ⊗ ∆ − ) ◦ ∆ − ,as well as the coassociativity property for ∆ − s , multiplicativity of ∆ + and ∆ + s withrespect to the tree product, and multiplicativity of ∆ − and ∆ − s with respect to theforest product. Recall that the label set L splits as L = L + ∪ L − , where L − indexes the set of“noises” while L + indexes the set of kernels, which in the setting of [BCCH17]equivalently indexes the components of the class of SPDEs under consideration.It is then natural to introduce another space assignment called a target spaceassignment ( W t ) t ∈ L where, for each t ∈ L , the vector space W t is the target spacefor the corresponding noise or component of the solution. As we already sawin the discussion at the start of Section 5.1, given a noise taking values in somespace W t for some t ∈ L − , it is natural to assign to it a subspace of the regularitystructure that is isomorphic to the (algebraic) dual space W ∗ t . Then, for fixing thespace assignment ( V t ) t ∈ L used in the category theoretic constructions earlier in thissection, this motivates space assignments of the form V t def = (cid:26) W ∗ t for t ∈ L − , R for t ∈ L + . (5.23)Given a space assignment V of the form (5.23), we then use the functor F V definedin Section 5.2 to define the vector spaces T , T + , T − , F , F − by U def = F V ( U ) = Y τ ∈ U ( R ) U [ τ ] , U [ τ ] def = F V ( h τ i ) = V ⊗h τ i (5.24)where, respectively, • U is one of T , T + , T − , F , F − , • U is one of T , T + , T − , F , F − , and • U is one of T , T + , T − , F , F − ( τ in (5.24) can denote either a tree or a forest).We also adopt a similar notation for linear maps h : U → X (for any vector space X ) by writing h [ τ ] for the restriction of h to U [ τ ] for any τ ∈ U ( R ) . Note thatthe forest product turns F and F − into algebras, but that T (or T + , T − ) are notalgebras in general since they may not be preserved by the tree product. We call T the concrete regularity structure built from T . The notion of sectors of T isdefined as before in [Hai14, Def. 2.5]. egularity structures for vector-valued noises Given a label decomposition ¯ L of L under p , we will naturally“extend” p to a map p : E → P ( ¯ E ) , where ¯ E = ¯ L × N d +1 , by setting, for o = ( t , p ) ∈ E , p ( o ) = p ( t ) × { p } .If we have a splitting of labels L = L + ⊔ L − then we implicitly work with acorresponding splitting ¯ L = ¯ L + ⊔ ¯ L − given by ¯ L ± = F t ∈ L ± p ( t ) .Additionally, given a rule R with respect to the labelling set L , we obtain acorresponding rule ¯ R with respect to ¯ L by setting, for each ¯ t ∈ ¯ L , ¯ R ( ¯ t ) = n ¯ N ∈ ˆ P ( E ) : ∃ N ∈ R ( t ) with ¯ N N o ,where t is the unique element of L with ¯ t ∈ p ( t ) and we say that ¯ N N if thereis a bijection from ¯ N to N respecting p . If we are also given a notion of degree deg : L → R then we also have an induced degree deg : ¯ L → R by setting, for ¯ t and t as above, deg( ¯ t ) = t . It then follows that subcriticality or completeness holdfor ¯ R if and only if they hold for R Linking back to Remark 5.31, we also mention that { p ( τ ) : τ ∈ T L ( R ) } is apartition of T ¯ L ( ¯ R ) .We say that a decomposition p acts trivially on l ∈ L if | p ( l ) | = 1 and in this casewe will just write p ( l ) = { l } . Remark 5.36
We call a decomposition ¯ L of L under p that acts trivially on L + a noise decomposition . The regularity structure T defined in (5.24) is then fixed, up tonatural transformation, under noise decompositions of L . In this case, if ( ¯ W l ) l ∈ ¯ L isa decomposition of the original target space assignment ( W t ) t ∈ L then ¯ V = ( ¯ V l ) l ∈ ¯ L built from the target space assignment ( ¯ W l ) l ∈ ¯ L using (5.23) is a decomposition of ( V t ) t ∈ L . Note that this is not the case if the decomposition acts non-trivially onelements of L + . In what follows, we will only directly apply the framework of thissubsection to handle noise decompositions. Remark 5.37
We call a target space assignment ( W t ) t ∈ L with dim( W t ) = 1 forevery t ∈ L − a scalar noise target space assignment, and we will say that we areworking with scalar noises. Our construction of regularity structures and renor-malisation groups in this section will match the constructions in [Hai14, BHZ19]when we have a scalar noise target space assignment and so we will have all themachinery developed in [Hai14, CH16, BHZ19, BCCH17] available.Given a set of labels L and target space assignment ( W t ) t ∈ L , we say ¯ L and ( ¯ W l ) l ∈ ¯ L are a scalar noise decomposition of L and ( W t ) t ∈ L if ¯ L is a noise decom-position and if ( ¯ W l ) l ∈ ¯ L is scalar noise target space assignment. In this situationnatural transformations given by Proposition 5.28 allow us to identify the regu-larity structure and renormalisation group built from L and ( W t ) t ∈ L with thosebuilt from ¯ L and ( ¯ W l ) l ∈ ¯ L . Thanks to this we can leverage the machinery of Respecting p means that if ¯ N ∋ ¯ o o ∈ N then ¯ o ∈ p ( o ) where p : E → P ( ¯ E ) as above. cf. (5.13) (note that a target space assignment is also a space assignment) egularity structures for vector-valued noises [Hai14, CH16, BHZ19, BCCH17] for the regularity structure and renormalisationgroup built from L and ( W t ) t ∈ L . Remark 5.38
One remaining difference between the setting of a scalar noise targetspace assignment and the setting of [Hai14, BHZ19] is that in [Hai14, BHZ19] onealso enforces dim( W t ) = 1 for t ∈ L + – this constraint enforces solutions to alsobe scalar-valued. However, while a scalar noise assignment allows dim( W t ) > for t ∈ L + , our decision to enforce V t = R in (5.23) means that we require that the“integration” encoded by edges of type t acts diagonally on W t , i.e., it doesn’t mixcomponents. In particular, with this constraint the difference between working withvector-valued solutions versus the corresponding system of equations with scalarsolutions is completely cosmetic – the underlying regularity structures are the sameand the only difference is how one organises the space of modelled distributions. Remark 5.39
While the convention (5.23) is natural in our setting, an examplewhere it must be discarded is the setting of [GH19]. In [GH19] combinatorialtrees also index subspaces of the regularity structure which generically are notone-dimensional. To start translating [GH19] into our framework one would wantto take L + = { t + } and set V t + = B for B an appropriate space of distributions.However, since B is infinite-dimensional in this case, the machinery we developin the remainder of this section does not immediately extend to this context. Seehowever [GHM20] for a trick allowing to circumvent this in some cases.At this point we make the following assumption. Assumption 5.40
Our target space assignments W are always finite-dimensionalspace assignments (which means the corresponding V given by (5.23) are finite-dimensional). Remark 5.41
There are several ways in which we use Assumption 5.40 in therest of this section. One key fact is that for vector spaces X and Y one has L ( X, Y ) ≃ X ∗ ⊗ Y provided that either X or Y is finite-dimensional – this is es-pecially important in the context of Remark 5.48. Another convenience of workingwith finite-dimensional space assignments is that we are then allowed to assumethe existence of a scalar noise decomposition which lets us leverage the machineryof [Hai14, CH16, BHZ19, BCCH17]. Our construction also provides us with a “renormalisation group” that remains fixedunder noise decompositions. Recalling the set of forests F − ( R ) and the associatedalgebra F − introduced in Section 5.6 (in particular Eq. (5.24)), we note that themap ∆ − s introduced in Section 5.5 – or rather its image under the functor F V –turns F − into a bialgebra. Moreover, we can use the number of edges of eachelement in F − ( R ) to grade F − . Since F − is connected by (5.22) (i.e. its subspaceof degree is generated by the unit), it admits an antipode A − turning it into a egularity structures for vector-valued noises commutative Hopf algebra and we denote by G − the associated group of characters.It is immediate that, up to natural isomorphisms, the Hopf algebra F − and charactergroup G − remain fixed under noise decompositions. Given ℓ ∈ G − , we define acorresponding renormalisation operator M ℓ , which is a linear operator M ℓ : T → T , M ℓ def = ( ℓ ⊗ id T ) ∆ − . (5.25) Remark 5.42
Note that the action of M ℓ would not in general be well defined onthe direct product F V ( T ) but it is well-defined on T thanks to the assumption ofsubcriticality.Also note that there is a canonical isomorphism G − ≃ M τ ∈ T − ( R ) T [ τ ] ∗ . (5.26)In particular, given ℓ ∈ G − and τ ∈ T − ( R ) , we write ℓ [ τ ] for the component of ℓ in T [ τ ] ∗ above. For the remainder of this section we impose the following assumption.
Assumption 5.43
The rule R satisfies R ( l ) = { () } for every l ∈ L − . A kernel assignment is a collection of kernels K = ( K t : t ∈ L + ) where each K t is a smooth compactly supported scalar function on R d +1 \ { } . A smooth noiseassignment is a tuple ζ = ( ζ t : t ∈ L − ) where each ζ t is a smooth function from R d +1 to W t .Note that the set of kernel (or smooth noise) assignments for L and W canbe identified with the set of kernel (or smooth noise) assignments for any ¯ L and ¯ W obtained via noise decomposition of the label set L and W . For smooth noiseassignments this identification is given by the correspondence ( ζ l : l ∈ L − ) ↔ ( ζ ¯ l = P ¯ l ζ l : l ∈ L − , ¯ l ∈ p ( l ) ) . If we are working with scalar noises then, upon fixing kernel and smooth noiseassignments K and ζ , [Hai14] introduces a map Π can which takes trees τ ∈ T ( R ) into C ∞ ( R d +1 ) . This map gives a correspondence between combinatorial treesand the space-time functions/distributions they represent (without incorporatingany negative or positive renormalisation), and Π can is extended linearly to T .In the general case with vector valued noise we can appeal to any scalar noisedecomposition ¯ L of L and W to again obtain a linear map Π can : T ¯ L → C ∞ ( R d +1 ) –this map is of course is independent of the particular scalar noise decompositionwe appealed to for its definition. egularity structures for vector-valued noises In order to make combinatorial arguments which use the structure of the treesof our abstract regularity structure, it is convenient to have an explicit vectorialformula for Π can .Given τ ∈ T ( R ) , we write h L ( τ ) i for the symmetric set obtained by restrictingthe tree symmetries of every τ ∈ τ to the set L ( τ ) def = { e ∈ E τ : t ( e ) ∈ L − } . Recalling the choice (5.23), we have W ⊗h L ( τ ) i ≃ ( V ∗ ) ⊗h τ i ≃ T [ τ ] ∗ , (5.27)where we used Assumption 5.40 and (5.10) for the second canonical isomorphism.Writing, for any τ ∈ T ( R ) , Π can [ τ ] for the restriction of Π can to T [ τ ] , we willrealise Π can [ τ ] as an element Π can [ τ ] ∈ C ∞ ( R d +1 , W ⊗h L ( τ ) i ) ,where we remind the reader that (5.27) gives us C ∞ ( R d +1 , W ⊗h L ( τ ) i ) ≃ L ( T [ τ ] , C ∞ ( R d +1 )) . The explicit vectorial formula for Π can mentioned above is then given by Π can [ τ ]( z ) = Z ( R d +1 ) N ( τ ) d x N ( τ ) δ ( x ̺ − z ) (cid:16) Y v ∈ N ( τ ) x n ( v ) v (cid:17) (5.28) (cid:16) Y e ∈ K ( τ ) D n ( e ) K t ( e ) ( x e + − x e − ) (cid:17)(cid:16) O e ∈ L ( τ ) D n ( e ) ζ t ( e ) ( x e + ) (cid:17) where we have taken an arbitrary τ ∈ τ and set K ( τ ) def = E τ \ L ( τ ) and N ( τ ) def = { v ∈ V τ : v = e − for any e ∈ L ( τ ) } . (5.29)Thanks to Assumption 5.43, all the integration variables x v ∈ R d +1 appearing onthe right-hand side of (5.28) satisfy v ∈ N ( τ ) . Moreover, while the right-hand sideof (5.28) is written as an element of N e ∈ L ( τ ) W t ( e ) , due to the symmetry of theintegrand it can canonically be identified with an element of W ⊗h L ( τ ) i . In the scalar noise setting, upon fixing a kernel assignment K and a random smooth noise assignment ζ , [BHZ19, Sec. 6.3] introduces a multiplicative linearfunctional ¯ Π can (denoted by g − ( Π can ) therein) on F − obtained by setting, for τ ∈ T ( R ) , ¯ Π can [ τ ] = E ( Π can [ τ ]( )) and then extending multiplicatively and linearly to With appropriate finite moment conditions. egularity structures for vector-valued noises the algebra F − . [BHZ19] also introduces a corresponding BPHZ renormalisationcharacter ℓ bphz ∈ G − given by ℓ bphz = ¯ Π can ◦ ˜ A − , (5.30)where ˜ A − is the negative twisted antipode , an algebra homomorphism from F − to F determined by enforcing the condition that M ( ˜ A − ⊗ id) ∆ − = 0 (5.31)on the subspace of T generated by T − ( R ) . Here, M denotes the (forest) multiplica-tion map from F ⊗ T into F . We remind the reader that condition (5.31), combinedwith multiplicativity of ˜ A − , gives a recursive method for computing ˜ A − τ wherethe recursion is in | E τ | .In the general vector-valued case we note that, analogously to (5.31), we canconsider for a morphism ˜ A − ∈ Hom( F − , F ) the identity M ( ˜ A − ⊗ id) ∆ − = 0 (5.32)as an identity in Hom( T − , F ) . If we furthermore impose that ˜ A − is multiplicativein the sense that ˜ A − ◦ M = M ◦ ( ˜ A − ⊗ ˜ A − ) as morphisms in Hom( F − ⊗ F − , F ) and ˜ A − [ ] = id F [ ] , then we can proceed again by induction on | E τ | to uniquelydetermine ˜ A − ∈ Hom( F − , F ) .Analogously to (5.27), one has T [ f ] ∗ ≃ W ⊗h L ( f ) i where h L ( f ) i is the sym-metric set obtained by restricting the forest symmetries of every f ∈ f to the setof leaves L ( f ) = S τ ∈ f L ( τ ) , where the union runs over all the trees τ in f . Thisshows that, if we set again ¯ Π can [ τ ] = E ( Π can [ τ ]( ) ) ,with Π can given in (5.28), we can view ¯ Π can [ τ ] as an element of T [ τ ] ∗ , and,extending its definition multiplicatively, as an element of T [ f ] ∗ . Hence (5.30)yields again an element of G − , provided that we set ˜ A − = F V ( ˜ A − ) . Remark 5.44
This construction is consistent with [BHZ19] in the sense that if weconsider ℓ bphz as in [BHZ19] for any scalar noise decomposition p of L , then thisagrees with the construction we just described, provided that the correspondingspaces are identified via the functor p ∗ . Υ In this subsection we will use type decompositions to show that the formula for the Υ map which appears in the description of the renormalisation of systems of scalarequations in [BCCH17] has an analogue in our setting of vector-valued regularitystructures. We again fix a finite label set L def = L + ⊔ L − and a target space assignment ( W t ) t ∈ L , which determines the space assignment ( V t ) t ∈ L by (5.23). Recall that denotes the empty forest and is the unit of the algebras F and F − , while is thetree with a single vertex and is just the zero of a vector space, so these three notations are completelydifferent. egularity structures for vector-valued noises Up to now we were consistently working with isomorphism classes τ ∈ T . For brevity, we will henceforth work with concrete trees τ ∈ τ , allconsiderations for which will depend only on the symmetric set h τ i , which, byRemark 5.29, we canonically identify with h τ i . We will correspondingly abusenotation and write τ ∈ T . Remark 5.46
In what follows we will often identify L with a subset of E = L × N d +1 by associating t ( t , ) .We define A def = Q o ∈E W o where the ( W ( t ,k ) : k ∈ N d +1 ) are distinct copies of thespace W t . One should think of A ∈ A as describing the jet of both the noise andthe solution to a system of PDEs of the form (5.1). We equip A with the producttopology.Given any two topological vector spaces U and B , we write C ∞ ( U, B ) for thespace of all maps F : U → B with the property that, for every element ℓ ∈ B ∗ , thereexists a continuous linear map ¯ ℓ : U → R n and a smooth function F ℓ, ¯ ℓ : R n → R such that, for every u ∈ U , h ℓ, F ( u ) i = F ℓ, ¯ ℓ (¯ ℓ ( u ) ) . (5.33)When our domain is U = A we often just write C ∞ ( B ) instead of C ∞ ( A , B ) . Notethat when B is finite-dimensional then for each F ∈ C ∞ ( B ) , F ( A ) is a smoothfunction of ( A o : o ∈ E F ) for some finite subset E F ⊂ E . Remark 5.47
One difference in the point of view of the present article versus thatof [BCCH17] is that here we will treat the solution and noise on a more equalfooting. As an example, in [BCCH17] the domain of our smooth functions wouldbe a direct product indexed by L + × N d +1 rather than one indexed by E . Thefact that, in the case of the stochastic Yang–Mills equations considered here, thedependence on the noise variables has to be affine is enforced when assuming thatthe nonlinearity obeys our rule R , see Definition 5.51 below.In particular, when defining the Υ map in [BCCH17] through an inductionon trees τ , the symbols associated to the noises (and derivatives and productsthereof) were treated as “generators” – the base case of the Υ induction. In oursetting, however, the sole such generator will be the symbol and noises will betreated as branches / edges I t ( ) for t ∈ L − . See also Remark 5.56 below.A specification of the right-hand side of our equation determines an element in ˚ Q def = C ∞ (cid:16) M t ∈ L ( V t ⊗ W t ) (cid:17) ≃ M t ∈ L C ∞ ( V t ⊗ W t ) . (5.34)Recalling that V t = R for t ∈ L + (see (5.23)) and writing an element F ∈ ˚ Q as F = L t ∈ L F t with F t ∈ C ∞ ( V t ⊗ W t ) , note that F t for t ∈ L + plays the role of Here we are referring to the “drivers” of [BCCH17]. egularity structures for vector-valued noises the function appearing on the right-hand side of (5.1), namely a smooth functionin the variable A ∈ A taking values in V t ⊗ W t ≃ W t . In our case, the space V t = R for t ∈ L + plays not much of a role, but in general it can be used to encodeadditional information about the integration kernel as in [GH19]. Remark 5.48
While the definition of the vector space A depends on the label set L and target space assignment ( W t ) t ∈ L , it is natural to treat A as remaining fixedunder decompositions of L and ( W t ) t ∈ L since the direct product that defines A would just re-sum our decomposition.Since we chose to simply set V t = R for t ∈ L + , the spaces C ∞ ( V t ⊗ W t ) with t ∈ L + are also invariant (modulo canonical isomorphisms) under decompositionsof L . This is not the case for these spaces with t ∈ L − . Indeed, given a finite-dimensional vector space B , the identity id B is the unique (up to multiplicationby a scalar) element of L ( B, B ) ≃ B ∗ ⊗ B such that, for every decomposition B = L i B i one has id B ∈ L i L ( B i , B i ) ≃ L i ( B ∗ i ⊗ B i ) . This suggests that ifwe want to have nonlinearities that are invariant under decomposition (and constantfor l ∈ L − as enforced by Assumption 5.43), we should set F l = id W l for l ∈ L − .This is indeed the case and will be enforced in Definition 5.52 below. Just as in [BCCH17] we introduce two families of differentiation operators, the first { D o } o ∈E corresponding to derivatives with respect to the components of the jet A and the second { ∂ j } dj =0 corresponding to derivatives in the underlying space-time.Consider locally convex topological vector spaces U and B . Suppose that B = Q i ∈ I B i , where each B i is finite-dimensional, equipped with the producttopology. Let F ∈ C ∞ ( U, B ) and ℓ ∈ B ∗ , ¯ ℓ , and F ℓ, ¯ ℓ as in (5.33). For m ≥ and u ∈ U , consider the symmetric m -linear map U m ∋ ( v , . . . , v m ) D m F ℓ, ¯ ℓ ( ¯ ℓ ( u ))( ¯ ℓ ( v ) , . . . , ¯ ℓ ( v m )) ∈ R . (5.35)For fixed u, v , . . . , v m , the right-hand side of (5.35) defines a linear function of ℓ which one can verify is independent of the choice of ¯ ℓ . Since B ∗ = L i ∈ I B ∗ i , thealgebraic dual of which is again B , there exists an element D m F ( u )( v , . . . , v m ) ∈ B such that h ℓ, D m F ( u )( v , . . . , v m ) i agrees with the right-hand side of (5.35). Itis immediate that U m ∋ ( v , . . . , v m ) D m F ( u )( v , . . . , v m ) ∈ B is symmetricand m -linear for every u ∈ U .Turning to the case U = A , for o , . . . , o m ∈ E and A ∈ A , we define D o · · · D o m F ( A ) = D m F ( A ) ↾ W o × ... × W om ∈ L ( W o , . . . , W o m ; B ) . Due to the finite-dimensionality of W o i by Assumption 5.40, the map D o · · · D o m F : A D o · · · D o m F ( A ) The ( A , ξ ) written in (5.1) corresponds to the A here, see Remark 5.47. egularity structures for vector-valued noises is an element of C ∞ ( L ( W o , . . . , W o m ; B )) . The operators { D o } o ∈E naturallycommute, modulo reordering the corresponding factors. Remark 5.49
Given a decomposition of our labelling set L and target space assign-ment ( W t ) t ∈ L into ¯ L and ( ¯ W t ) t ∈ ¯ L , our definitions give us another set of derivativeoperators { D ¯ o } ¯ o ∈ ¯ E . Via the identifications W o = L ¯ o ∈ p ( o ) W ¯ o for any o ∈ E , wehave for any F ∈ C ∞ ( B ) D o F ≃ M ¯ o ∈ p ( o ) D ¯ o F , (5.36)where p ( o ) is understood as in Remark 5.35. Analogous identities involving iterateddirect sums hold for iterated derivatives.For j ∈ { , , . . . , d } and o = ( t , p ) ∈ E , we first define ∂ j o ∈ E by ∂ j o = ( t , p + e j ) .We then define operators ∂ j on C ∞ ( B ) by setting, for A = ( A o ) o ∈E ∈ A and F ∈ C ∞ ( B ) , ( ∂ j F ) ( A ) def = X o ∈E ( D o F ) ( A ) A ∂ j o . (5.37)Note that the operators { ∂ j } dj =0 commute amongst themselves and so ∂ p is well-defined for any p ∈ N d +1 . Remark 5.50
Combining (5.36) with (5.37), we see that the definition of thederivatives { ∂ j } dj =0 remains unchanged under decompositions of L and ( W t ) t ∈ L ,thus justifying our notation.Just as in [BCCH17], we want to restrict ourselves to F ∈ ˚ Q that obey the rule R we use to construct our regularity structure. Definition 5.51
We say F ∈ ˚ Q obeys a rule R if, for each t ∈ L and o , . . . , o n ∈ E , ( o , · · · , o n ) R ( t ) ⇒ D o · · · D o n F t = 0 . (5.38)Note that, for any type decomposition ¯ L of L under p , F obeys a rule R if and onlyif it obeys ¯ R as defined in Remark 5.35. Definition 5.52
Given a subcritical and complete rule R , define Q ⊂ ˚ Q to be theset of F obeying R such that furthermore F l ( A ) = id W l for all l ∈ L − .Recall that, by Assumption 5.43, for any F obeying the rule R , F l ( A ) is independentof A for l ∈ L − . The reason for imposing the specific choice F l ( A ) = id W l is furtherdiscussed in Remarks 5.48 and 5.60 below. The notion of obey (and the set Q ) we choose here is analogous to item (ii) of [BCCH17,Prop. 3.13] rather than [BCCH17, Def. 3.10]. In particular, our definition is not based on expandinga nonlinearity in terms of a polynomial in the rough components of A and a smooth function in theregular components of A . This will means we don’t need to impose [BCCH17, Assump. 3.12] to usethe main results of [BCCH17]. egularity structures for vector-valued noises Υ In this subsection we formulate the notion of coherence from [BCCH17, Sec. 3] inthe setting of vector regularity structures. In particular, in Theorem 5.59, we showthat the coherence constraint is preserved under noise decompositions.We first introduce some useful notation. For o = ( t , p ) ∈ E we set B def = F V ( h T i ) = Y τ ∈ T V ⊗h τ i , B o def = F V ( h I o T i ) ⊂ B ,and equip B with the product topology. As usual, we use the notation B t = B ( t , ) .Note that B is an algebra when equipped with the tree product, or rather itsimage under F V . The following remark, where we should have in mind the case B + = F V ( h T \ { }i ) , is crucial for the formulation of our construction. Remark 5.53
Let B + be an algebra such that B + = lim ←− n B ( n ) + with each B ( n ) + nilpotent, let U and B be locally convex spaces where B is of the same form as inSection 5.8.1, and let F ∈ C ∞ ( U, B ) . Write B = R ⊕ B + , which is then a unitalalgebra. Then F can be extended to a map B ⊗ U → B ⊗ B as follows: for u ∈ U and ˜ u ∈ B + ⊗ U , we set F ( ⊗ u + ˜ u ) def = X m ∈ N D m F ( u ) m ! ( ˜ u, . . . , ˜ u ) , (5.39)where D m F ( u ) ∈ L ( U, . . . , U ; B ) for each u ∈ U naturally extends to a m -linearmap ( B ⊗ U ) m → B ⊗ B by imposing that D m F ( u )( b ⊗ v , . . . , b m ⊗ v m ) = ( b · · · b m ) ⊗ D m F ( u )( v , . . . , v m ) . (5.40)Note that the first term of this series (5.39) belongs to R ⊗ B while all other termbelong to B + ⊗ B . Since ˜ u ∈ B + ⊗ U , the projection of this series onto any of thespaces B ( n ) + ⊗ B contains only finitely many non-zero terms by nilpotency, so thatit is guaranteed to converge. If B and all the B ( n ) + are finite-dimensional, then theextension of F defined in (5.39) actually belongs to C ∞ ( B ⊗ U, B ⊗ B ) , where B + is equipped with the projective limit topology, under which it is nuclear, and B ⊗ U , is equipped with the projective tensor product. Furthermore, in this case,every element of C ∞ ( B ⊗ U, B ⊗ B ) extends to an element of C ∞ ( B ˆ ⊗ U, B ⊗ B ) ,where ˆ ⊗ denotes the (completion of the) projective tensor product.We introduce a space of expansions H = L t ∈ L H t with H t def = ( B t ⊕ ¯ T ) ⊗ W t ⊂ B ⊗ W t , ¯ T def = Y k ∈ N d +1 T [ X k ] . The space H introduced here plays the role of the space H ex in [BCCH17, Sec. 3.7]. egularity structures for vector-valued noises Given
A ∈ H , we write A = P t ∈ L A t with A t = A R t + (cid:16) X k ∈ N d +1 X k k ! ⊗ A ( t ,k ) (cid:17) ∈ H t , A R t ∈ B t ⊗ W t , (5.41)and write A A = ( A o ) o ∈E ∈ A , where the coefficients A ( t ,k ) are as in (5.41). Notethat (5.41) gives a natural inclusion A ⊂ H . Remark 5.54
In (5.41) and several places to follow, we write P to denote anelement of a direct product. This will simplify several expressions below, e.g. (5.53).For o = ( t , p ) ∈ E we also define A o def = A Ro + (cid:16) X k ∈ N d +1 X k k ! ⊗ A ( t ,p + k ) (cid:17) ∈ B ⊗ W o , (5.42)where A Ro ∈ B o ⊗ W o is given by the image of A R t under the canonical isomorphism B o ⊗ W o ≃ B t ⊗ W t (but note that B o and B t are different subspaces of B when p = 0 ). Collecting the A o into one element ˆ A def = ( A o : o ∈ E ) , we see that ˆ A isnaturally viewed as an element of B ˆ ⊗ A .Note also that, for any o = ( t , p ) ∈ E , our construction of the morphism I o onsymmetric sets built from trees in Section 5.4.1 gives us, via the functor F V , anisomorphism I o : B ⊗ V t → B o .We fix for the rest of this subsection a choice of F ∈ Q . Then the statementthat A ∈ H algebraically solves (5.1) corresponds to A R t = ( I t ⊗ id W t ) ( F t ( ˆ A ) ) . (5.43)Here, we used Remark 5.53 to view F t : A → V t ⊗ W t as a map from B ˆ ⊗ A into B ⊗ V t ⊗ W t , which I t then maps into B t ⊗ W t .The coherence condition then encodes the constraint (5.43) as a functionaldependence of A R = L t ∈ L A R t on A A . This functional dependence will beformulated by defining a pair of (essentially equivalent) maps Υ and ¯ Υ where ¯ Υ ∈ Y t ∈ L C ∞ ( B ⊗ V t ⊗ W t ) and Υ ∈ Y o ∈E C ∞ ( B o ⊗ W o ) . (5.44)The coherence condition on A ∈ H will be formulated as A R = Υ ( A A ) .To define ¯ Υ and Υ , we first define corresponding maps ¯Υ and Υ (belongingto the same respective spaces) from which ¯ Υ and Υ will be obtained by includingsome combinatorial factors (see (5.52)). We will write, for t ∈ L , o ∈ E , and τ ∈ T , Υ o [ τ ] for the component of Υ in C ∞ ( B [ I o τ ] ⊗ W o ) and ¯Υ t [ τ ] for its component As a component of A A ∈ A , A o ∈ W o , while as a term of (5.41), A o ∈ W t . This is of coursenot a problem since W o ≃ W t . egularity structures for vector-valued noises in C ∞ ( B [ τ ] ⊗ V t ⊗ W t ) . We will define Υ and ¯Υ by specifying the components Υ o [ τ ] and ¯Υ t [ τ ] through an induction in τ . Before describing this induction, wemake another remark about notation. Remark 5.55
For t ∈ L , G ∈ C ∞ ( V t ⊗ W t ) , ( o , τ ) , . . . , ( o m , τ m ) ∈ E × T , and k ∈ N d +1 , consider the map ∂ k D o · · · D o m G ∈ C ∞ ( L ( W o , . . . , W o m ; V t ⊗ W t )) . It follows from Remark 5.53 that if we are given elements Θ i ∈ C ∞ ( B ⊗ W o i ) wehave a canonical interpretation for ( ∂ k D o · · · D o m G ( A ) ) ( Θ ( A ) , . . . , Θ m ( A )) ∈ B ⊗ V t ⊗ W t , (5.45)which, as a function of A , is an element of C ∞ ( B ⊗ V t ⊗ W t ) which we denote by ( ∂ k D o · · · D o m G ) ( Θ , . . . , Θ m ) .We further note that if Θ i ∈ C ∞ ( B i ⊗ W o i ) for some subspaces B i ⊂ B , then(5.45) belongs to ˆ B ⊗ V t ⊗ W t , where ˆ B ⊂ B is the smallest closed linear spacecontaining all products of the form b · · · b m with b i ∈ B i .Now consider an isomorphism class of trees τ ∈ T . Then τ can be written as X k m Y i =1 I o i ( τ i ) , (5.46)where k = n ( ̺ ) , m ≥ , τ i ∈ T , and o i ∈ E . Remark 5.56
Following up on Remark 5.47, in the analogous expression [BCCH17,Eq. (2.11)] a tree τ may also contain a factor Ξ representing a noise. However, in(5.46) a noise (or a derivative of a noise) is represented by I ( l ,p ) ( ) with l ∈ L − .Given t ∈ L and τ of the form (5.46), ¯Υ t [ τ ] and Υ ( t ,p ) [ τ ] are inductively definedby first setting ¯Υ t [ ] def = ⊗ F t , (5.47)which belongs to B [ ] ⊗ C ∞ ( V t ⊗ W t ) ≃ C ∞ ( B [ ] ⊗ V t ⊗ W t ) ⊂ C ∞ ( B ⊗ V t ⊗ W t ) (the first isomorphism follows from the fact that V t ⊗ W t is finite dimensional byassumption) so this is indeed of the desired type. We then set ¯Υ t [ τ ] def = X k h ∂ k D o · · · D o m ¯Υ t [ ] i (Υ o [ τ ] , . . . , Υ o m [ τ m ] ) , Υ ( t ,p ) [ τ ] def = ( I ( t ,p ) ⊗ id W t ) ( ¯Υ t [ τ ]) . (5.48) This means that our notation for Υ [ τ ] breaks the notational convention we’ve used so far forother elements of spaces of this type (direct products of F V ( h τ i ) , possibly tensorised with some fixedspace). The reason we do this is to be compatible with the notations of [BCCH17], and also to keepnotations in Sections 6.2 and 7.3 cleaner. egularity structures for vector-valued noises We explain some of the notation and conventions used in (5.48). By Remark 5.55,the term following X k in the right-hand side for ¯Υ t [ τ ] is an element of C ∞ (cid:16) B h m Y j =1 I o j ( τ j ) i ⊗ V t ⊗ W t (cid:17) . We then interpret X k • as the canonical isomorphism B h m Y j =1 I o j ( τ j ) i ≃ B h X k m Y j =1 I o j ( τ j ) i = B [ τ ] , (5.49)acting on the first factor of the tensor product, hence the right-hand side ofthe definition of ¯Υ t [ τ ] belongs to C ∞ ( B [ τ ] ⊗ V t ⊗ W t ) , which is mapped to C ∞ ( B [ I ( t ,p ) τ ] ⊗ W t ) by I ( t ,p ) ⊗ id W t as desired. Remark 5.57
We have two important consequences of (5.48) and (5.47):(i) Since F obeys R , we have, for any o = ( t , p ) ∈ E , Υ o [ τ ] = 0 and ¯Υ t [ τ ] = 0 unless I t ( τ ) ∈ T ( R ) .(ii) For any t ∈ L − , p ∈ N d +1 and τ ∈ T \ { } , one has Υ ( t ,p ) [ τ ] = 0 and ¯Υ t [ τ ] = 0 , due to annihilation by the operators ∂ and D .In particular, our assumption that the rule R is subcritical guarantees that for anygiven degree γ , only finitely many of the components ¯Υ t [ τ ] with deg τ < γ arenon-vanishing. Remark 5.58
Although it plays exactly the same role, the map Υ introduced in[BCCH17] is of a slightly different type than the maps Υ and ¯Υ introduced here.More precisely, in [BCCH17], Υ o [ τ ]( A ) ∈ R played the role of a coefficient ofa basis vector in the regularity structure. In the present article on the other hand, Υ o [ τ ]( A ) ∈ B [ I o τ ] ⊗ W o . In the setting of [BCCH17], these spaces are canonicallyisomorphic to R and our definitions are consistent modulo this isomorphism.The ¯Υ and Υ defined in (5.48) are missing the combinatorial symmetry factors S ( τ ) associated to a tree τ ∈ T , which we define in the same way as in [BCCH17]. Forthis we represent τ more explicitly than (5.46) by writing τ = X k ℓ Y j =1 I o j ( τ j ) β j , (5.50)with ℓ ≥ , β j > , and distinct ( o , τ ) , . . . , ( o ℓ , τ ℓ ) ∈ E × T , and define S ( τ ) def = k ! (cid:16) ℓ Y j =1 S ( τ j ) β j β j ! (cid:17) . (5.51) The coefficient of I o ( τ ) in a coherent jet and the coefficient of τ in the expansion of thenon-linearities evaluated on a coherent jet. egularity structures for vector-valued noises We then set, for t ∈ L , o ∈ E , and τ ∈ T , Υ = X o ∈E ,τ ∈ T Υ o [ τ ] , Υ o [ τ ] def = Υ o [ τ ] /S ( τ ) , ¯Υ = X t ∈ L ,τ ∈ T ¯Υ t [ τ ] , ¯Υ t [ τ ] def = ¯Υ t [ τ ] /S ( τ ) . (5.52)We can now state the main theorem of this section – here we specialise to noisedecompositions (i.e. those acting trivially on L + ) described in Remark 5.36. Theorem 5.59 ¯Υ and Υ as defined in (5.52) are left unchanged under noisedecompositions of L and ( W t ) t ∈ L . Precisely, given a noise decomposition ¯ L of L under p with associated target space decomposition ( ¯ W l ) l ∈ ¯ L , one has, for any t ∈ L , τ ∈ T and A ∈ A , Υ t [ τ ]( A ) = X l ∈ p ( t ) , ¯ τ ∈ p ( τ ) Υ l [ ¯ τ ]( A ) , (5.53) where ( Υ l ) l ∈ ¯ L on the right-hand side is defined as above but with the decomposedlabelling set and target space assignment used in its construction, and p ( τ ) isdefined as in Remark 5.31. The equality (5.53) also holds when Υ is replaced by ¯Υ .Proof. We prove (5.53) inductively in the number of edges of τ . Writing τ in theform (5.46), our base case corresponds to m = 0 , i.e. τ = X k for k ∈ N d +1 , sothat p ( X k ) = { X k } and S ( X k ) = k ! . This case is covered by Remark 5.48. For ourinductive step, we may assume that m ≥ in (5.46). By item (ii) of Remark 5.57there is nothing to check if t ∈ L − so we turn to the situation where t ∈ L + .Since p acts trivially on L + we can write p ( t ) = { t } . Then, inserting ourinductive hypothesis in (5.48) and also applying Remark 5.49 we see that (5.53)follows if we can show that X ( l,σ ) ∈ d ( τ ) ( I t ⊗ id W t ) h X k (cid:16) ∂ k ( D l · · · D l m ¯Υ t [ ] )(Υ l [ σ ] , . . . , Υ l m [ σ m ] ) (cid:17)i = X ¯ τ ∈ p ( τ ) S ( τ ) S ( ¯ τ ) Υ t [ ¯ τ ] , (5.54)where d ( τ ) consists of all pairs of tuples ( l, σ ) with l = ( l i ) mi =1 , σ = ( σ i ) mi =1 , l i ∈ p ( o i ) and σ i ∈ p ( τ i ) . Given ( l, σ ) ∈ d ( τ ) we write τ ( l, σ ) def = X k Q mi =1 I l i ( σ i ) ∈ T ¯ L .Clearly one has τ ( l, σ ) ∈ p ( τ ) and for fixed ( l, σ ) the corresponding summand onthe left-hand side of (5.54) is simply Υ t [ τ ( l, σ )] . Finally, for any ¯ τ ∈ p ( τ ) , it isstraightforward to prove, using a simple induction and manipulations of multinomialcoefficients, that S ( τ ) S ( ¯ τ ) = |{ ( l, σ ) ∈ d ( τ ) : τ ( l, σ ) = ¯ τ }| , egularity structures for vector-valued noises which shows (5.54). Using natural isomorphisms between the spaces where ¯Υ and Υ live, it follows that (5.53) also holds for ¯Υ . Remark 5.60
We used Remark 5.48 in a crucial way to start the induction, whichshows that since we consider rules with R ( l ) = { () } for l ∈ L − , the choice F l = id W l is the only one that complies with (5.38) and also guarantees invariance under noisedecompositions. See however [GHM20] for an example where R ( l ) = { () } and itis natural to make a different choice for F l .We now precisely define coherence in our setting. For L ∈ N ∪ {∞} , we denoteby p ≤ L the projection map on L t ∈ L B ⊗ W t which vanishes on any subspace ofthe form T [ τ ] ⊗ W t if | E τ | + X v ∈ V τ | n ( v ) | > L and is the identity otherwise. Above we write E τ for the set of edges of τ , V τ forthe set of nodes of τ , and n for the label on τ . Note that p ≤∞ is just the identityoperator. Definition 5.61
We say
A ∈ H is coherent to order L ∈ N ∪ {∞} with F if p ≤ L A R = p ≤ L Υ ( A A ) , (5.55)where A R = L t ∈ L A R t with A R t determined from A as in (5.41).Note that, by Theorem 5.59, coherence to any order L is preserved under noise de-compositions. Thanks to Theorem 5.59 we can reformulate [BCCH17, Lem. 3.21]to show that our definition of Υ encodes the condition (5.43); we state this as alemma. Lemma 5.62
A ∈ H is coherent to order L ∈ N ∪ {∞} with F if and only if, foreach t ∈ L , p ≤ L A R t = p ≤ L ( I t ⊗ id W t ) F t ( A ) . (5.56) Remark 5.63
Combining Lemma 5.62 with Definition 5.61 shows that Υ doesindeed have the advertised property, namely it yields a formula for the “non-standardpart” of the expansion of any solution to the algebraic counterpart (5.43) of the mildformulation of the original problem (5.1).Conversely, this provides us with an alternative method for computing Υ ( A )[ τ ] for any τ ∈ T . Given A ∈ A , set A ( ) = A ∈ H (recall (5.41) for the identificationof A as a subspace of H ) and then proceed iteratively by setting A ( n +1 ) t = A t + ( I t ⊗ id W t ) F t ( A ( n ) ) . Subcriticality then guarantees that any of the projections A ( n ) [ τ ] stabilises after afinite number of steps, and one has Υ t [ τ ]( A ) = A ( ∞ ) [ τ ] . egularity structures for vector-valued noises The material discussed in Section 5.8 up to this point has beendevoted to treating (5.1) as an algebraic fixed point problem in the space H . We alsowant to solve an analytic fixed point problem in a space of modelled distributions,namely in a space of H -valued functions over some space-time domain.Posing the analytic fixed point problem requires us to start with more input thanwe needed for the algebraic one. After fixing F ∈ Q one also needs to fix suitable regularity exponents ( γ t : t ∈ L ) for the modelled distribution spaces involved andinitial data ( u t : t ∈ L + ) for the problem. Moreover, one prescribes a modelleddistribution expansion for each noise, namely for every l ∈ L − , we fix a modelleddistribution O l of regularity D γ l of the form O l ( z ) = X k ∈ N d +1 O ( l ,k ) ( z ) X k + I l ( ) . (5.57)The corresponding analytic fixed point problem [BCCH17, Eq. (5.6)] is then posedon a space of modelled distributions U = ( U t : t ∈ L + ) such that U t ∈ D γ t (at leastlocally). On some space-time domain D (typically of the form [ , T ] × R d ), thefixed point problem is of the form U t = P t t> F t ( ( U ⊔ O )( • ) ) + G t u t . In this identity, P t is an operator of the form ( P t F ) ( z ) = p ≤ γ t I t F ( z ) + ( . . . ) ,where ( . . . ) takes values in ¯ T t def = L k ∈ N d +1 T [ X k ] ⊗ W t , and G t is the “harmonicextension map” as in [Hai14, (7.13)] associated to ( ∂ t − L t ) − (possibly withsuitable boundary conditions). Here, p ≤ γ t is the projection onto components ofdegree less than γ t . Since G t also takes values in ¯ T t , it follows that for any solution U to such a fixed point problem and any space-time point z ∈ D , U ( z ) ⊔ O ( z ) iscoherent with F to some order L which depends on the exponents ( γ t : t ∈ L ) ;see [BCCH17, Thm. 5.7] for a precise statement.Note that, depending on the degrees of our noises, there can be some freedom inour choice of (5.57) depending on how we choose to have our model act on symbols I l ( ) for l ∈ L − – the key fact is that O l represents the corresponding driving noisein our problem, not necessarily I l ( ) . However, when deg( l ) < , a natural choicefor the input (5.57) is to simply set O ( l ,k ) ( z ) = 0 for all k ∈ N d +1 and this is theconvention we use in Sections 6 and 7. We now describe the action of the renormalisation group G − on nonlinearities,which is how it produces counterterms in equations. We no longer treat F ∈ Q as Here, “suitable” means sufficiently large so that the fixed point problem is well-posed. Subcriti-cality guarantees that setting γ t = γ for all t is a suitable choice provided that γ ∈ R + is sufficientlylarge. olution theory of the SYM equation fixed and when we want to make the dependence of ¯ Υ on F ∈ Q explicit we write ¯ Υ F . We re-formulate the main algebraic results of [BCCH17] in the followingproposition; the proof is obtained by using Theorem 5.59 to restate [BCCH17,Lem. 3.22, Lem. 3.23, and Prop. 3.24]. Proposition 5.65
Fix ℓ ∈ G − . There is a map F M ℓ F , taking Q to itself,defined by, for t ∈ L and A ∈ A , ( M ℓ F ) t ( A ) def = ( p , t M ℓ ⊗ id V t ⊗ W t ) ¯ Υ F t ( A ) = F t ( A ) + X τ ∈ T − ( R ) ( ℓ ⊗ id V t ⊗ W t ) ¯ Υ F t [ τ ]( A ) , (5.58) where p , t denotes the projection onto T [ ] and the operator M ℓ on the right-handside is given by (5.25) .Moreover, for any L ∈ N ∪ {∞} , p ≤ L (cid:16) M t ∈ L M ℓ ⊗ id V t ⊗ W t (cid:17) ¯ Υ F = p ≤ L ¯ Υ M ℓ F , (5.59) and there exists ¯ L ∈ N ∪ {∞} (which can be taken finite if L is finite) such that if A ∈ H is coherent to order ¯ L with F ∈ Q then ( M ℓ ⊗ id) A is coherent to order L with M ℓ F . Remark 5.66
Note that for t ∈ L , ¯ Υ F t [ τ ]( A ) ∈ T [ τ ] ⊗ V t ⊗ W t , so every term onthe right-hand side of (5.58) is an element of V t ⊗ W t . In this section we make rigorous the solution theory for (1.7) and provide the proofof Theorem 2.4. In particular, we explicitly identify the counterterms appearing inthe renormalised equation as this will be needed for the proof of gauge covariancein Section 7. Recalling Remark 2.7 we make the following assumption.
Assumption 6.1
The Lie algebra g is simple. The current section is split into two parts. In Section 6.1 we recast (1.7) into theframework of regularity structures with vector-valued noise using Section 5. InSection 6.2 we invoke the black box theory of [Hai14, CH16, BHZ19, BCCH17]to prove convergence of our mollified / renormalised solutions and then explicitlycompute our renormalised equation (using Proposition 5.65) in order to show that,when d = 2 , the one counterterm appearing converges to a finite value as ε ↓ . olution theory of the SYM equation We set up our regularity structure for formulating (1.7) in d = 2 or d = 3 spacedimensions. However, when, computing counterterms, we will again fix d = 2 .We write [ d ] def = { , · · · , d } .Our space-time scaling s ∈ [ , ∞ ) d +1 is given by setting s = 2 and s i = 1 for i ∈ [ d ] . We define L + def = { a i } di =1 and L − def = { l i } di =1 . We define a degree deg : L → R on our label set by setting deg( t ) def = ( t ∈ L + , − d/ − − κ t ∈ L − (6.1)where we fix κ ∈ ( , / ) .Looking at equation (1.8) leads us to consider the rule ˚ R given by setting, foreach i ∈ [ d ] , ˚ R ( l i ) def = {6 } and ˚ R ( a i ) def = (cid:26) { ( l i , ) } , { ( a i , ) , ( a j , ) , ( a j , ) }{ ( a j , ) , ( a j , e i ) } , { ( a j , ) , ( a i , e j ) } : j ∈ [ d ] (cid:27) . (6.2)It is straightforward to verify that ˚ R is subcritical. The rule ˚ R has a smallest normal[BHZ19, Definition 5.22] extension and this extension admits a completion R asconstructed in [BHZ19, Proposition 5.21] – this rule R is also subcritical will bethe rule that will be used to define our regularity structure.We fix our target space assignment ( W t ) t ∈ L by setting W t def = g ∀ t ∈ L . (6.3)The space assignment ( V t ) t ∈ L used in the construction of our concrete regularitystructure via the functor F V is then given by (5.23). Remark 6.2
While the notation A = ( A ( t ,p ) : ( t , p ) ∈ E ) ∈ A was convenientfor the formulation and proof of the statements of Section 5.8, it would make thecomputations of this section and Section 7 harder to follow. We thus go back tousing the symbol A for the components A ( t ,p ) of A with t ∈ L + and the symbol ξ for the components A ( l , ) with l ∈ L − . To streamline notations, we also write thesubscript p as a derivative, namely we write ξ i = A ( l i , ) , A i = A ( a i , ) and ∂ j A i = A ( a i ,e j ) . (6.4)Regarding the specification of the right-hand side F = L t ∈ L F t ∈ Q , we set,and for each i ∈ [ d ] and A ∈ A , F l i ( A ) = id g and F a i ( A ) = A ( l i , ) + d X j =1 [ A ( a j , ) , A ( a i ,e j ) − A ( a j ,e i ) + [ A ( a j , ) , A ( a i , ) ]] = ξ i + d X j =1 [ A j , ∂ j A i − ∂ i A j + [ A j , A i ]] , (6.5) olution theory of the SYM equation where the identification of the two lines uses the notations of Remark 6.2.The right-hand side of (6.5) is clearly a polynomial in A taking values in W a i ≃ g as described in Section 5.8, so it is indeed the case that F ∈ ˚ Q . Thederivatives D o · · · D o m F a i ( A ) , for o , . . . , o m ∈ E , are not difficult to compute.For instance, for fixed A ∈ A , D ( a j , ) F a i ( A ) ∈ L ( W ( a j , ) , W ( a i , ) ) ≃ L ( g , g ) isgiven by ( D ( a j , ) F a i ( A ) ) ( • ) = [ • , ∂ j A i − ∂ i A j + [ A j , A i ]] + [ A j , [ • , A i ]] + δ i,j d X k =1 [ A k , [ A k , • ]] . It is then straightforward to see that F ∈ Q , namely, it obeys the rule R in the senseof Definition 5.51. d = 2 We remind the reader that we now restrict to the case case d = 2 and | l | s = − − κ for every l ∈ L − . We also enforce that κ < / .As mentioned in Remark 5.56, we will use the symbol Ξ i def = I ( l i , ) ( ) for thenoise. Similarly to Remark 6.2, we also use the notations I i and I i,j as shorthandsfor I ( a i , ) and I ( a i ,e j ) respectively.Below we introduce a graphical notation to describe forms of relevant trees.The noises Ξ i are circles , noises with polynomials X e j Ξ with j ∈ [ d ] are crossedcircles , and edges I i and I i,j are thin and thick grey lines respectively. It isalways assumed that the indices i and j appearing on occurrences of Ξ i , I i , and I i,j throughout the tree are constrained by the requirement that our trees conformto the rule R .We now give a complete list of the forms of trees in T ( R ) with negative degree,the form is listed on the top and the degree below it. , X k Ξ i , | k | s = 2 − − κ − − κ − − κ − κ − κ − κ Note that each symbol above actually corresponds to a family of trees, determinedby assigning indices in a way that conforms to the rule R . For instance, when wesay that τ is of the form , then τ could be any tree of the type I i ( I i ( Ξ i ) I i ,j ( Ξ i ) ) I i ,j ( Ξ i ) for any i , i , i , i , j , j ∈ [ d ] satisfying both of the following two constraints:first, one must have either i = i or j = i , and, second, one must have either i = i and j = i or i = j and i = i .Note that a circle or a crossed circle actually represents an edge when wethink of any of the corresponding typed combinatorial trees; for instance, in thesense of Section 5.4, has four edges and not two. In Section 6.2.4, we willfurther colour our graphical symbols to encode constraints on indices. olution theory of the SYM equation (1.7)We fix a kernel assignment by setting, for every t ∈ L + , K t = K where we fix K to be a truncation of the Green’s function G ( z ) of the heat operator which satisfiesthe following properties:1. K ( z ) is smooth on R d +1 \ { } .2. K ( z ) = G ( z ) for ≤ | z | s ≤ / .3. K ( z ) = 0 for | z | s >
4. Writing z = ( t, x ) with x ∈ T , K ( , x ) = 0 for x = 0 , and K ( t, x ) = 0 for t < .5. Writing z = ( t, x , x ) with x , x the spatial components, K ( t, x , x ) = K ( t, − x , x ) = K ( t, x , − x ) and K ( t, x , x ) = K ( t, x , x ) .We will also use the shorthand K ε = K ∗ χ ε . Remark 6.3
Property 4 is not strictly necessary for the proof of Theorem 2.1 butwill be convenient for proving item (ii) of Theorem 2.8 in Section 7 so we includeit here for convenience. Property 5 is also not strictly necessary, but convenient ifwe want certain BPHZ renormalisation constants to vanish rather than just beingfinite.Note that we do not assume a moment vanishing condition here as in [Hai14,Assumption 5.4] – the only real change from the framework of [Hai14] that droppingthis assumption entails is that, for p, k ∈ N d +1 , we can have presence of expressionssuch as I ( m ,p ) ( X k ) when we write out trees in T ( R ) . Works such as [CH16],[BHZ19], [BCCH17] already assume trees containing such expressions are allowedto be present.Next, we overload notation and introduce a random smooth noise assignment ζ ε = ( ζ l ) l ∈ L − by setting, for i ∈ [ d ] , ζ l ( i ) = ξ εi where we recall that ξ εi = ξ i ∗ χ ε and ( ξ i ) di =1 are the i.i.d. g -valued space-time white noises introduced as the beginningof the paper. With this fixed choice of kernel assignment and random smooth noiseassignment ζ ε for ε > we have a corresponding BPHZ renormalised model Z ε bphz .We also write ℓ ε bphz ∈ G − for the corresponding BPHZ character. (1.7)We now apply [CH16, Theorem 2.15] to prove the following. Lemma 6.4
The random models Z ε bphz converge in probability, as ε ↓ , to alimiting random model Z bphz .Proof. We note that [CH16, Theorem 2.15] is stated for the scalar noise setting soto be precise one must verify its conditions after applying some choice of scalarnoise decomposition. However, it is not hard to see that the conditions of thetheorem are completely insensitive to the choice of scalar noise decomposition. Let ζ = ( ζ l ) l ∈ L − be the unmollified random noise assignment, that is, ζ l ( i ) = ξ i . olution theory of the SYM equation For any scalar noise decomposition, it is straightforward to verify the conditionthat the random smooth noise assignments ζ ε are a uniformly compatible familyof Gaussian noises that converge to the Gaussian noise ζ (again, seen as a rough,random, noise assignment for scalar noise decomposition in the natural way).The first three listed conditions of [CH16, Theorem 2.15] refer to power-counting considerations written in terms of the degrees of the combinatorial treesspanning the scalar regularity structure and the degrees of the noises. Since thispower-counting is not affected by decompositions, they can be checked directly onthe trees of T ( R ) . We note that min { deg( τ ) : τ ∈ T ( R ) , | N ( τ ) | > } = − − κ > − is achieved for τ of the form – here N ( τ ) is as defined in (5.29). This is greaterthan −| s | / , so the third criterion is satisfied. Combining this with the fact that deg( l ) = −| s | / − κ for every l ∈ L − guarantees that the second criterion issatisfied. Finally, the worst case scenario for the first condition is for τ of the formand A = { a } with type t ( a ) = l for any element l ∈ L − and for which we have deg( τ ) + deg( ¯ l ) + | s | = 2 − κ > ,as required. The set of trees T − ( R ) is given by all trees of the form , , , , , , , , or . Our remaining objective for this section is to compute the counterterms X τ ∈ T − ( R ) ( ℓ ε bphz [ τ ] ⊗ id W t ) ¯ Υ F t [ τ ]( A ) (6.6)for each t ∈ L + . In what follows, we perform separate computations for thecharacter ℓ ε bphz [ τ ] and for ¯ Υ F , before combining them to compute (6.6). Thefollowing lemma identifies some cases where ℓ ε bphz [ τ ] = 0 . Lemma 6.5 (i) ℓ ε bphz [ τ ] = 0 for each τ consisting of an odd number of noises,that is any τ of the form , , and .(ii) On every subspace T [ τ ] of the regularity structure with τ of the form , ,or , one has ˜ A − = − id , so that ℓ ε bphz [ τ ] = − ¯ Π can [ τ ] .(iii) ℓ ε bphz [ τ ] = 0 for every τ of the form .(iv) For τ of the form , or , one has ℓ ε bphz [ τ ] = 0 unless the two noises Ξ i and Ξ i appearing in τ carry the same index, that is i = i .(v) For τ of the form or , one has ℓ ε bphz [ τ ] = 0 unless the two spatialderivatives appearing on the two thick edges in τ carry the same index. olution theory of the SYM equation Proof.
Item (i) is true for every Gaussian noise. For item (ii), the statement aboutthe abstract regularity structure is a direct consequence of the definition (5.32) ofthe twisted antipode (see also [BHZ19, Prop. 6.6]) and the statement about ℓ ε bphz [ τ ] then follows from (5.30).For item (iii) if we write τ = I i ( Ξ i ) I j,l ( Ξ j ) then ¯ Π can [ τ ] = Z du dv K ( − u ) ∂ l K ( − v ) E [ ξ εi ( u ) ⊗ ξ εj ( v )] . Performing a change of variable by flipping the sign of the l -component of v ,followed by exploiting the equality in law of ξ ε and the change in sign of ∂ l K undersuch a reflection, shows that the integral above vanishes.For item (iv), the fact that E [ ξ εi ( u ) ⊗ ξ εj ( v )] = 0 if i = j enforces the desiredconstraint.For item (v), the argument is similar to that of item (iii) - namely the presence ofprecisely one spatial derivative in a given direction allows one to argue that ¯ Π can [ τ ] vanishes by performing a reflection in the appropriate integration variable in thatdirection. Remark 6.6
We now start to use splotches of colour such as or to representindices in [ d ] , since they will allow us to work with expressions that would becomeunwieldy when using Greek or Roman letters. We also use Kronecker notation toenforce the equality of indices represented by colours, for instance writing δ , .We can use colours to include indices in our graphical notation for trees in anunobtrusive way, for instance writing = I ( ) = I ( Ξ ) . Note that the splotchof in the symbol fixes the two indices in I ( Ξ ) which have to be equal forany tree conforming to our rule R . The edges corresponding to integration can bedecorated by derivatives which introduce a new index, so we introduce notationsuch as = I , ( Ξ ) , where the colour of a thick edge determines the index of itsderivative.For drawing a tree like I ( Ξ ) I , ( I , ( Ξ )) , our earlier way of drawingdidn’t give us a node to colour , so we add small triangular nodes to our drawingsto allow us to display the colour determining the type of the edge incident to thatnode from below, for example = I ( Ξ ) I , ( I , ( Ξ )) .We will see by Lemma 6.13 that Υ i [ τ ] = 0 for any τ of the form or .Therefore, (6.6) will only have contributions from trees of the form, , or . (6.7)Thanks to the invariance of our driving noises under the action of the Lie groupwe will see in Lemma 6.9 below that ℓ ε bphz [ τ ] has to be a scalar multiple of theCasimir element (in particular, it belongs to the subspace g ⊗ s g ⊂ g ⊗ g ). Thisis an immediate consequence of using noise that is white with respect to our innerproduct h· , ·i on g . In particular, note that this inner product on g induces an inner olution theory of the SYM equation product h· , ·i on g ⊗ g and that there is a unique element Cas ∈ g ⊗ s g ⊂ g ⊗ g with the property that, for any h , h ∈ g , h Cas , h ⊗ h i = h h , h i . (6.8)One can write Cas explicitly as
Cas = P i e i ⊗ e i for any orthonormal basis of g but we will refrain from doing so since we want to perform computations withoutfixing a basis. Cas should be thought of as the covariance of g -valued white noise,in particular for i, j ∈ { , } we have E [ ξ i ( t, x ) ⊗ ξ j ( s, y )] = δ i,j δ ( t − s ) δ ( x − y ) Cas . (6.9)Thanks to (6.8), Cas is invariant under the action of the Lie group G in the sensethat (Ad g ⊗ Ad g ) Cas = Cas , ∀ g ∈ G . (6.10)The identity (6.10) is of course just a statement about the rotation invariance ofour noise. Alternatively, we can interpret
Cas as an element of U ( g ) , the universalenveloping algebra of g . The following standard fact will be crucial in the sequel. Lemma 6.7
Cas belongs to the centre of U ( g ) .Proof. Let h ∈ g and let θ be a random element of g with E ( θ ⊗ θ ) = Cas .Differentiating E [Ad g θ ⊗ Ad g θ ] at g = e in the direction of h yields E ( [ h, θ ] ⊗ θ ) = − E ( θ ⊗ [ h, θ ] ) . (6.11)We conclude that [ h, Cas] = [ h, E ( θ ⊗ θ )] = E ( h ⊗ θ ⊗ θ − θ ⊗ θ ⊗ h ) = E ( [ h, θ ] ⊗ θ + θ ⊗ [ h, θ ] ) = 0 ,as claimed, where we used (6.11) in the last step. Remark 6.8
We note that
Cas is of course just the quadratic Casimir. Moreover,recall that every element h ∈ U ( g ) yields a linear operator ad h : g → g by setting ad h ⊗···⊗ h k X = [ h , · · · [ h k , X ] · · · ] . With this notation, Lemma 6.7 implies that ad Cas commutes with every other oper-ator of the form ad h for h ∈ g (and therefore also h ∈ U ( g ) ). If g is simple, thenthis implies that ad Cas = λ id g .We now describe ℓ ε bphz [ τ ] for τ of the form (6.7). We define ¯ C ε def = Z dz K ε ( z ) , ˆ C ε def = Z dz ∂ j K ε ( z )( ∂ j K ∗ K ε )( z ) , C ε sym def = ¯ C ε − C ε (6.12)where, on the right-hand side of the second equation one can choose either j = 1 or – they both give the same value. olution theory of the SYM equation For ˆ C ε and ¯ C ε as in (6.12) , one has ℓ ε bphz [ ] = − ℓ ε bphz [ ] = ˆ C ε Cas , ℓ ε bphz [ ] = ¯ C ε Cas . (6.13) Furthermore, C ε sym as defined in (6.12) converges to a finite value C sym as ε → . Remark 6.10
The last statement is specific to two space dimensions. In threedimensions, the mass renormalisations do not add up to a finite constant.
Remark 6.11
The first identity of (6.13) makes sense since, even though there aretwo natural isomorphisms T [ ] ∗ ≈ g ⊗ g and T [ ] ∗ ≈ g ⊗ g (corresponding tothe two ways of matching the two noises), Cas is invariant under that transposition.For the second identity, note that ℓ ε bphz [ ] ≃ g ⊗ s g . Proof.
The identities (6.13) readily follow from direct computation once one usesthat in all cases ℓ ε bphz [ τ ] = − ¯ Π can [ τ ] (this is item (ii) of Lemma 6.5), writes downthe corresponding expectation / integral, moves the mollification from the noises tothe kernels, and uses (6.9).Regarding the last claim of the lemma, since K is a truncation of the heat kernel,observe that ( ∂ t − ∆ ) K = δ + Q , (6.14)where Q is smooth and supported away from the origin. Using the shorthand Int[ F ] = R dz F ( z ) it follows that Int[( ∆ K ∗ K ε ) K ε ] = − Int[( K ε ) ] + Int[( ∂ t K ∗ K ε ) K ε ] − Int[( Q ∗ K ε ) K ε ] . (6.15)On the other hand, we also have Int[( ∆ K ∗ K ε ) K ε ] = Int[( K ∗ K ε ) ∆ K ε ] (6.16) = − Int[( K ∗ K ε ) χ ε ] − Int[( ∂ t K ∗ K ε ) K ε ] − Int[( K ∗ K ε )( Q ∗ χ ε )] . Observe that
Int[( K ∗ K ε ) χ ε ] , Int[( Q ∗ K ε ) K ε ] , and Int[( K ∗ K ε )( Q ∗ χ ε )] allconverge as ε → . Hence, adding (6.15) and (6.16), we obtain that Int[( ∆ K ∗ K ε ) K ε ] + Int[( K ε ) ] (6.17) = − Int[( K ∗ K ε ) χ ε ] − Int[( K ∗ K ε )( Q ∗ χ ε )] − Int[( Q ∗ K ε ) K ε ] converges as ε → . We now note that the quantity above equals C ε sym since ¯ C ε = Int[( K ε ) ] and, by integration by parts, ˆ C ε = − Int[( ∆ K ∗ K ε ) K ε ] ,which completes the proof. These facts follow easily from the fact that K ∗ K is well-defined and bounded and also continuousaway from the origin. To see this note that the semigroup property gives ( G ∗ G )( t, x ) = tG ( t, x ) and G − K is smooth and supported away from the origin. olution theory of the SYM equation ¯ Υ F Before continuing, we introduce some notational conventions that will be convenientwhen we calculate ¯ Υ F .Recall that, for ∈ [ d ] , the symbol Ξ is a tree that indexes a subspace T [ Ξ ] of our concrete regularity structure T . We introduce a corresponding notation Ξ ∈ T [ Ξ ] ⊗ g which, under the isomorphism T [ Ξ ] ⊗ g ≃ g ∗ ⊗ g ≃ L ( g , g ) , isgiven by Ξ = id g . The expression Ξ really represents the corresponding noise inthe sense that ( Π can Ξ )( · ) = ξ ε ( · ) , where we are abusing notation by having Π can only act on the left factor of the tensor product.Continuing to develop this notation, we also define Ψ = I Ξ ∈ T [ ] ⊗ g . where we continue the same notation abuse, namely I acts only on the left factorsappearing in Ξ . In particular, we have Π can Ψ ( · ) = ( K ∗ ξ ε )( · ) . We also have acorresponding notation Ψ , = I , Ξ ∈ T [ ] ⊗ g . We now show how this notation is used for products / non-linear expressions. Givensome h ∈ g , we may write an expression such has [ Ψ , [ h, Ψ , ]] ∈ T [ ] ⊗ g . (6.18)In an expression like this, we apply the multiplication T [ ] ⊗ T [ ] → T [ ] tocombine the left factors of Ψ and Ψ , . The right g -factors of Ψ and Ψ , areused as the actual arguments of the brackets above, yielding the new g -factor on theright. Remark 6.12
For what follows, given i, j ∈ [ d ] , we write Υ i and Υ i,j for Υ a i and Υ ( a i ,e j ) respectively. In particular, we will use notation such as Υ and Υ , . Weextend this convention, also writing Υ and ¯ Υ along with W and W , .With these conventions in place the following computations follow quite easily fromour definitions: ¯Υ F [ ] = δ , Ξ , Υ F [ ] = δ , Ψ , Υ F, [ ] = δ , Ψ , . (6.19)(Since S ( τ ) = 1 for these trees, the corresponding Υ are identical.) Moreover, for k ∈ N d +1 with k = 0 , Υ F [ X k Ξ ] = Υ F, [ X k Ξ ] = 0 . (6.20)Note that the left-hand sides of (6.19) are in principle allowed to depend on anargument A ∈ A , but here they are constant in A , so we are using a canonicalidentification of constants with constant functions here. We now compute ¯Υ F for all the trees appearing in (6.7). In (6.19) we are also exploiting canonical isomorphisms between W and W , and g . Forinstance, one also has ¯Υ F, [ ] = δ , Ξ but here the last g factor on the right-hand side should beinterpreted via the isomorphism with W , rather than W as in the first equality of (6.19). olution theory of the SYM equation ¯ Υ F [ ]( A ) = = [ Ψ , [ Ψ , A ]] (6.21) ¯ Υ F [ ]( A ) = ( δ , δ , − δ , δ , ) [[ δ , A − δ , A , I ( Ψ , )] , Ψ , ] ¯ Υ F [ ]( A ) = ( δ , δ , − δ , δ , )[ Ψ , [ δ , A − δ , A , I , ( Ψ , )]] Moreover, for any τ of the form or , ¯ Υ F t [ τ ] = 0 for every t ∈ L + .Proof. Let τ = I ( τ ) I ( τ ) for some choice of trees τ and τ , one has, by (5.48), ¯Υ F [ τ ]( A ) = = = (cid:16) [ Υ F [ τ ]( A ) , [ Υ F [ τ ]( A ) , A ]] + [ Υ F [ τ ]( A ) , [ Υ F [ τ ]( A ) , A ]] (cid:17) + = = (cid:16) [ Υ F [ τ ]( A ) , [ A , Υ F [ τ ]( A )]] + [ A , [ Υ F [ τ ]( A ) , Υ F [ τ ]( A )]] (cid:17) . Specifying to τ = , using (6.19) in the above identity gives ¯Υ F [ ]( A ) = = = (cid:16) [ Ψ , [ Ψ , A ]] + [ Ψ , [ Ψ , A ]] (cid:17) + = = (cid:16) [ Ψ , [ A , Ψ ]] + [ A , [ Ψ , Ψ ]] (cid:17) . (6.22)Therefore, ¯Υ F [ ]( A ) = 2 = [ Ψ , [ Ψ , A ]] . (6.23)By (5.51) we have S ( ) = ( ) S ( ) = 2 and so the first identity of (6.21) follows.Before moving onto the second identity we recall that, again using (5.48), ¯Υ F [ ]( A ) = [ δ , A − δ , A , Υ F, [ ]( A )] = [ δ , A − δ , A , Ψ , ] . (6.24)It follows that ¯Υ F [ ]( A ) = ( δ , δ , − δ , δ , )[ Υ F [ ]( A ) , Υ F, [ ]( A )] = ( δ , δ , − δ , δ , ) [[ δ , A − δ , A , I ( Ψ , )] , Ψ , ] where again, in I ( Ψ , ) ∈ T [ ] ⊗ g , the operator I is acting only on the left factorof Ψ , ∈ T [ ] ⊗ g . We then obtain the second identity of (6.21) since S ( ) = 1 .For the third identity we recall that, by (5.48) and (6.24), Υ F, [ ] = I , ( ¯Υ F [ ]) = [2 δ , A − δ , A , I , ( Ψ , ) ] , Note that = here is necessary, because different colours only means “not necessarily identical”! olution theory of the SYM equation so that ¯Υ F [ ]( A ) = ( δ , δ , − δ , δ , )[ Υ F [ ]( A ) , Υ F, [ ]( A )] = ( δ , δ , − δ , δ , ) [Ψ , [2 δ , A − δ , A , I , ( Ψ , ) ]] . Since S ( ) = 1 we obtain the desired result.The final claim of the lemma regarding trees of the form or followsimmediately from the induction (5.48) combined with (6.20). Before proceeding, we give more detail on how to use our notation for computations.We note that, given any w ∈ g ⊗ s g , one can use the isomorphism T [ ] ∗ ≃ g ⊗ g to view w as acting on the expression (6.18) via an adjoint action, namely, ( w ⊗ id g ) [Ψ , [ h, Ψ , ] ] = − ( w ⊗ id g ) [Ψ , [ Ψ , , h ] ] = − ad w h . (6.25) Lemma 6.14 X τ ∈ T − ( R ) ( ℓ ε bphz [ τ ] ⊗ id g ) ¯ Υ F [ τ ]( A ) = λC ε sym A , (6.26) where λ is the constant given in Remark 6.8 and C ε sym is as in (6.12) .Proof. By (6.21) and Lemma 6.9, X , , ( ℓ ε bphz [ ] ⊗ id g ) ¯ Υ [ ]( A ) = ( ˆ C ε Cas ⊗ id g ) (cid:16) [[ A , I Ψ , ] , Ψ , ] − X [[ A , I Ψ , ] , Ψ , ] − X [[ A , I Ψ , ] , Ψ , ] + [[ A , I Ψ , ] , Ψ , ] (cid:17) = − C ε ad Cas A ,where we used d = 2 to sum over the free indices and , as well as (6.25) in thelast step.Also, by (6.21) and Lemma 6.9 X , , ( ℓ ε bphz [ ] ⊗ id g ) ¯ Υ [ ]( A ) = ( − ˆ C ε Cas ⊗ id g ) (cid:16) [ Ψ , [ A , I , Ψ , ]] − [ Ψ , [ A , I , Ψ , ]] Again, this is only canonical up to permutation of the factors, but doesn’t matter since w issymmetric. olution theory of the SYM equation − X [ Ψ , [ A , I , Ψ , ]] + [ Ψ , [ A , I , Ψ , ]] (cid:17) = − ˆ C ε ad Cas A ,where d = 2 and (6.25) are used again. Finally by (6.21) and Lemma 6.9, X ( ℓ ε bphz [ ] ⊗ id g ) ¯ Υ F [ ]( A ) = ¯ C ε ad Cas
A .
Adding these three terms and recalling Remark 6.8 gives (6.26).With these calculation, we are ready for the proof of Theorem 2.4.
Remark 6.15
It would be desirable to apply the black box convergence theo-rem [BCCH17, Thm. 2.21] directly. However, we are slightly outside its scopesince we are working with non-standard spaces Ω α and are required to show conti-nuity at time t = 0 for the solution A ε : [ , T ] → Ω α . Nonetheless, we can insteaduse several more general results from [BCCH17, BHZ19, CH16, Hai14]. Proof of Theorem 2.4.
Consider the lifted equation associated to (2.1) in the bundleof modelled distributions D γ,η − κ ⋉ M for γ > κ (and η as before). Note that γ > κ and η > − ensure that, by [Hai14, Thm. 7.8], the lifted equation admitsa unique fixed point A ∈ D γ,η − κ and is locally Lipschitz in ( a, Z ) ∈ Ω C η × M , where M is the space of models on the associated regularity structure. Specialising to Z = Z ε bphz , the computation of Lemma 6.14 along with [BCCH17, Thm. 5.7](and its partial reformulation in the vector case via Proposition 5.65) show that thereconstruction A ε def = RA ( a, Z ε bphz ) is the maximal solution in Ω C η to the PDE (2.1)starting from a with C replaced by C ε given by C ε = λC ε sym , (6.27)where C ε sym is as in Lemma 6.9. We now show that A ε converges in the space Ω sol .To this end, let us decompose A ε = Ψ ε + B ε , where Ψ ε solves ∂ t Ψ ε = ∆Ψ ε + ξ ε on R + × T with initial condition a ∈ Ω α . Write also Ψ for the solution to ∂ t Ψ = ∆Ψ+ ξ on R + × T with initial condition a ∈ Ω α . Combining Theorem 4.13and Proposition 4.6, we see that Ψ ∈ C ( R + , Ω α ) , and, by Corollary 4.14, Ψ ε → Ψ in C ( R + , Ω α ) . Moreover, observe that B ε = R ( P + F ( A )) , where F ( A ) ∈ D γ − − κ, η − − − κ . From the embedding Ω C α/ ֒ → Ω α (Remark 3.23), the convergenceof models given by Lemma 6.4, the continuity of the reconstruction map, and [Hai14,Thm. 7.1], we see that B ε converges in the space Ω sol as ε → .Finally, observe that we can perturb the constants C ε in (2.1) by any boundedquantity while retaining convergence of maximal solutions to (2.1) and so, thanksto the convergence of (6.27) promised by Lemma 6.9, we obtain the desired con-vergence for any family ( C ε ) ε ∈ ( , ] such that lim ε → C ε exists and is finite. auge covariance The proof of Theorem 2.4 allows us to make the following importantobservation about dependence of the solution on the mollifier. Namely, given anyfixed constant δC ∈ R , the limiting maximal solution to (2.1) obtained as one takes ε ↓ with C = δC + λC sym is independent of the choice of mollifier, namely alldependence of the solution on the mollifier is cancelled by C sym ’s dependence onthe mollifier. Moreover, recall that C sym is the ε ↓ limit of the right-hand side of (6.17) and lim ε ↓ Z dz ( K ∗ K ε )( z )( Q ∗ χ ε )( z ) and lim ε ↓ Z dz ( Q ∗ K ε )( z ) K ε ( z ) are both independent of χ , where Q is as in (6.14). In particular, if one chooses C = − λ lim ε ↓ Z dz χ ε ( z )( K ∗ K ε )( z ) then the limiting solution to (2.1) is independent of the mollifier χ . The aim of this section is to show that the projected process [ A t ] on the orbit spaceis again a Markov process, which is a very strong form of gauge invariance. The firstthree subsections will be devoted to proving Theorem 2.8 – most of our work willbe devoted to part (i) and we will obtain part (ii) afterwards by a short computationwith renormalisation constants. We close with Section 7.4 where we construct thedesired Markov process. One obstruction we encounter when trying to directly treat the systems (2.5) and(2.6) using currently available tools is that the evolution for g takes place in thenon-linear space G . Fortunately, it is possible to rewrite the equations in such a waythat the role of g is played by objects living in linear spaces. For this, given a smoothfunction g : T → G , we define the functions h : T → g and U : T → L ( g , g ) by h def = (d g ) g − and U def = Ad g . (7.1)Straightforward algebraic manipulations yield the following lemma. Lemma 7.1
Given solutions B and g to (2.5) and defining h and U by (7.1) , onehas ∂ t h i = ∆ h i − [ h j , ∂ j h i ] + [[ B j , h j ] , h i ] + ∂ i [ B j , h j ] , ∂ t U = ∆ U − [ h j , [ h j , · ]] ◦ U + [[ B j , h j ] , · ] ◦ U , ∂ t B i = ∆ B i + [ B j , ∂ j B i − ∂ i B j + [ B j , B i ]] + U J ε ( ξ i ) + CB i + Ch i . (7.2) This is because λC sym is the renormalisation arising from limiting BPHZ model Z bphz whichis independent of the mollifier. auge covariance Similarly, given solutions ¯ A and ¯ g to (2.6) and defining ¯ h and ¯ U by (7.1) in termsof ¯ g , one has ∂ t ¯ A i = ∆ ¯ A i + [ ¯ A j , ∂ j ¯ A i − ∂ i ¯ A j + [ ¯ A j , ¯ A i ]] + J ε ( ¯ U ξ i ) + C ¯ A i + ( C − ¯ C ) ¯ h i , (7.3) and the equations for ¯ h and ¯ U are as in (7.2) with B replaced by ¯ A . Note that in the above equations we have omitted the ε -dependence in the notation B, ¯ A, U, ¯ U , h, ¯ h , while ξ is the white noise which is really independent of ε . Theterm ¯ U ξ i appearing in (7.3) is well-defined since ¯ A i is smooth for any given ε > . Proof.
By definition (7.1) of h and the equation for g in (2.5), one has the followingidentities ( ∂ t g ) g − = div h + [ B j , h j ] , ∂ j h i − ∂ i h j = [ h j , h i ] , ∆ h i − ∂ i div h = ∂ j [ h j , h i ] , (7.4)where the last identity follows from the second. One then obtains ∂ t h i = [( ∂ t g ) g − , h i ] + ∂ i (( ∂ t g ) g − ) = [div h + [ B j , h j ] , h i ] + ∂ i div h + ∂ i [ B j , h j ] = ∆ h i − [ h j , ∂ j h i ] + [[ B j , h j ] , h i ] + ∂ i [ B j , h j ] . For the U equation, we start by noting that ∂ i U = [ h i , · ] ◦ U (7.5)and therefore ∆ U = [div h, · ] ◦ U + [ h i , [ h i , · ]] ◦ U . (7.6)By the first identity in (7.4) ∂ t U = [( ∂ t g ) g − , · ] ◦ U = [div h + [ B j , h j ] , · ] ◦ U , (7.7)and the claim follows from (7.6). The equations for ¯ A, ¯ h, ¯ U are derived in the sameway. Remark 7.2
Note that the knowledge of U and h is sufficient to describe the actionof the corresponding g on connections since A g = U A − h . This means that, forthe purpose of our argument, we never need to go back and recover the evolutionof g from that of U and h . The same can of course be said for ¯ g , ¯ U , and ¯ h . auge covariance To recast (7.2) in the language of regularity structures, we use the label sets L + def = { a i , h i , m i } di =1 ∪ { u } and L − def = { ¯ l i , l i } di =1 . Our approach is to work with one single regularity structure to study both systems(7.2) and (7.3), allowing us to compare their solutions at the abstract level ofmodelled distributions.Our particular choice of label sets and abstract non-linearities also involvessome pre-processing to allow us to use the machinery of Section 5.8.3 to obtain theform of our renormalised equation. The label h i indexes the solutions h i or ¯ h i , u indexes the solutions U or ¯ U , and l i indexes the noise ξ i (while ¯ l i indexes a noisemollified at scale ε , see below).The other labels are used to describe the B i equation within system (7.2) andthe ¯ A i equation within system (7.3). To explain our strategy, we first note that(ignoring for the moment the contribution coming from the initial condition) theequation for ¯ A can be written as the integral fixed point equation ¯ A i = G ∗ ( [ ¯ A j , ∂ j ¯ A i − ∂ i ¯ A j + [ ¯ A j , ¯ A i ]] + C ¯ A i + ( C − ¯ C ) ¯ h i ) + G ε ∗ ( ¯ U ξ i ) . where G ε = χ ε ∗ G . While this can be cast as an abstract fixed point problem atthe level of jets / modelled distributions, it does not quite fit into the framework ofSection 5.8.3 since it involves multiple kernels on the right-hand side. We can dealwith this problem by introducing a component m i to index a new component ofour solution that is only used to represent the term G ε ∗ ( ¯ U ξ i ) . The label a i thenrepresents the first term on the right-hand side above.Turning to the equation for B , the corresponding fixed point problem is B i = G ∗ ( [ B j , ∂ j B i − ∂ i B j + [ B j , B i ]] + CB i + Ch i + U J ε ( ξ i ) ) Note that we cannot combine the mollification operator J ε with a heat kernel, so weinstead we use the label ¯ l i to represent J ε ( ξ i ) which we treat, at a purely algebraiclevel, as a completely separate noise from ξ i .Turning to our space assignment ( W t ) t ∈ L we set W t def = ( g t = a i , h i , l i , ¯ l i , or m i , L ( g , g ) t = u , (7.8)and the space assignment ( V t ) t ∈ L is given by (5.23) as before. We also define deg : L → R by setting deg( t ) def = − κ t = a i or m i , t = h i or u , − d/ − − κ t = l i or ¯ l i ,where κ ∈ ( , / ) . auge covariance The systems of equations (7.2) and (7.3) and earlier discussion about the rolesof our labels lead us to the rule ˚ R given by setting ˚ R ( l i ) = ˚ R ( ¯ l i ) = { } , ˚ R ( m i ) = { ul i } , ˚ R ( u ) = { uh j , uq j h j : q ∈ { a , m } , j ∈ [ d ] } , ˚ R ( h i ) = { h j ∂ j h i , q j h j h i , h j ∂ i q j , q j ∂ i h j : q ∈ { a , m } , j ∈ [ d ] } , ˚ R ( a i ) = { q i , h i , q i ˆ q j ˜ q j , q j ∂ i ˆ q j , q j ∂ j ˆ q i , ul i , u ¯ l i : q , ˆ q , ˜ q ∈ { a , m } , j ∈ [ d ] } . (7.9)Here we are using monomial notation for node types: a type t ∈ L should beassociated with ( t , ) and the symbol ∂ j t represents ( t , e j ) . We write products torepresent multisets, for instance a j ∂ k a i = { ( a i , e k ) , ( a j , ) , ( a j , ) } . We write q , ˆ q ,and ˜ q as dummy symbols since any occurrence of B i or ¯ A i can correspond to anoccurrence of a i or m i .It is straightforward to check that ˚ R is subcritical and as in Section 6.1 therule ˚ R has a smallest normal extension which admits a completion R which is alsosubcritical. This is the rule that is used to define the set of trees T ( R ) which is usedto build our regularity structure.We adopt conventions analogous to those of Remark 6.2 and (6.4), writing(using our monomial notation) ¯ A i = A a i + A m i , ∂ j ¯ A i = A ∂ j a i + A ∂ j m i ,B i = A a i , ∂ j B i = A ∂ j a i , U = ¯ U = A u , ∂ i U = ∂ i ¯ U = A ∂ i u , J ε ( ξ i ) = ξ ¯ l i , ξ i = ξ l i , h i = ¯ h i = A h i , ∂ j h i = ∂ j ¯ h i = A ∂ j h i . (7.10)Here, we choose to typeset components of A in purple in order to be able to identifythem at a glance as a solution -dependent element. This will be convenient lateron when we manipulate expressions belonging to T ⊗ W for some vector space W (typically W = g or W = L ( g , g ) ), in which case purple variables are alwayselements of the second factor W . Note that, when referring to components of A = ( A o ) o ∈E the symbols ¯ U , ¯ h , and ∂ j ¯ h i are identical to their unbarred versionsbut we still use both notations depending on which system of equations we areworking with.We now fix two non-linearities F = L t ∈ L F t , ¯ F = L t ∈ L ¯ F t ∈ Q , whichencode our systems (7.2) and (7.3), respectively. For some constants ˚ C , and ˚ C to The choice to include { ul i } ∈ ˚ R ( a i ) isn’t directly motivated by (7.2) and (7.3) but is neededwhen we want to write an expression like (7.17) where H is the modelled distribution representing ¯ U . Components of A corresponding to the noise such as ξ l i and ξ ¯ l i will be left in black since theirvalues are not solution-dependent. auge covariance be fixed later we set F t and ¯ F t to be id g for t ∈ L − and F t ( A ) def = [ B j , ∂ j B i − ∂ i B j + [ B j , B i ]] + ˚ C B i + ˚ C h i + U J ε ( ξ i ) if t = a i , − [ h j , ∂ j h i ] + [[ B j , h j ] , h i ] + ∂ i [ B j , h j ] if t = h i , − [ h j , [ h j , · ]] ◦ U + [[ B j , h j ] , · ] ◦ U if t = u , if t = m i . (The term ∂ i [ B j , h j ] should be interpreted by formally applying the Leibniz rule.)For ¯ F t ( A ) we set ¯ F t ( A ) def = [ ¯ A j , ∂ j ¯ A i − ∂ i ¯ A j + [ ¯ A j , ¯ A i ]] + ˚ C ¯ A i + ˚ C ¯ h i if t = a i , − [ ¯ h j , ∂ j ¯ h i ] + [[ ¯ A j , ¯ h j ] , ¯ h i ] + ∂ i [ ¯ A j , ¯ h j ] if t = h i , − [ ¯ h j , [ ¯ h j , · ]] ◦ ¯ U + [[ ¯ A j , ¯ h j ] , · ] ◦ ¯ U if t = u , ¯ U ξ i if t = m i . For j ∈ { } ⊔ [ d ] we also introduce the shorthands Ξ i = I ( l i , ) ( ) , Ξ i = I ( ¯ l i , ) ( ) , I i,j ( · ) = I ( a i ,j ) ( · ) , ¯ I i,j ( · ) = I ( m i ,j ) ( · ) , I h i,j ( · ) = I ( h i ,j ) ( · ) , I u ( · ) = I ( u , ) ( · ) . When j = 0 in the above notation we sometimes suppress this index, for instancewriting I i ( · ) instead of I i, ( · ) . We write K ( ε ) = ( K ( ε ) t : t ∈ L + ) for the kernel assignment given by setting K ( ε ) t = (cid:26) K for t = a i , h i , or u , K ε = K ∗ χ ε for t = m i . (7.11)We also write M for the space of all models and, for ε ∈ [ , ] , we write M ε ⊂ M for the family of K ( ε ) -admissible models.Note that in our choice of degrees we enforced deg( a i ) = deg( m i ) = − − κ rather than − . The reason is that this allows us to extract a factor ε κ from anyoccurrence of K − K ε , which is crucial for Lemma 7.8, as well as the proof of(7.23) and (7.25) in Lemma 7.9.We make this more precise now. Recall first the notion of a β -regularisingkernel from [Hai14, Assumption 5.1]. We introduce some terminology so that wecan use that notion in a slightly more quantitative sense. For β, R > , r ≥ wesay that a kernel J is ( r, R, β ) -regularising, if one can find a decomposition of theform [Hai14, (5.3)] such that the estimates [Hai14, (5.4), (5.5)] hold with the samechoice of C = R for all multi-indices k, l with | k | s , | l | s < r . We use the norms One will see that these constants are shifts of the constant C by some finite constants that dependonly on our truncation of the heat kernel K . auge covariance ||| • ||| α,m on functions with prescribed singularities at the origin that were defined in[Hai14, Definition 10.12]. If J is a smooth function (except for possibly the origin),satisfies [Hai14, Assumption 5.4] with for some r ≥ , and is supported on the ball | x | ≤ , then it is straightforward to show that J is ( r, ||| J ||| β −| s | ,r , β ) -regularising.We then have the following key estimate. Lemma 7.3
For any m ∈ N one has ||| K ||| ,m < ∞ and there exists R such that K − K ε is ( m, ε κ R, − κ ) -regularising for all ε ∈ [ , ] .Proof. The first statement is standard. The second statement follows from combin-ing the first statement, our conditions on the kernel K , [Hai14, Lemma 10.17], andthe observations made above.We now turn to our random noise assignments. In (7.2) and (7.3) both a mollifiednoise J ε ( ξ i ) = ξ εi and an un-mollified noise ξ appear. In order to start our analysiswith smooth models, we replace the un-mollified noise with one mollified at scale δ . In particular, given ε, δ ∈ ( , ] we define a random noise assignment ζ δ,ε = ( ζ l : l ∈ L − ) by setting ζ l = (cid:26) χ δ ∗ ξ i = ξ δi for l = l i χ ε ∗ ξ i = ξ εi for l = ¯ l i .We also define Z δ,ε bphz = ( Π δ,ε , Γ δ,ε ) ∈ M ε to be the BPHZ lift associated to thekernel assignment K ( ε ) and random noise assignment ζ δ,ε . In our analysis we willwe will first take δ ↓ followed by ε ↓ – the first limit is a minor technical pointwhile the second limit is the limit referenced in part (i) of Theorem 2.8.Note that we have “doubled” our noises in our noise assignment by having twosets of noise labels { l i } di =1 and { ¯ l i } di =1 – we will want to use the fact that thesetwo sets of noises take values in the same space g (and in practice, differ only bymollification). This is formalised by noting that there are canonical isomorphisms T [ Ξ i ] ≃ g ∗ ≃ T [ ¯Ξ i ] for each i ∈ [ d ] , which we combine into an isomorphism σ : d M i =1 T [ Ξ i ] → d M i =1 T [ Ξ i ] . (7.12) ε -dependent regularity structures In the framework of regularity structures, analytic statements regarding models andmodelled distributions reference norms k • k ℓ on the vector space T ℓ of all elementsof degree ℓ ∈ deg( R ) = { deg( τ ) : τ ∈ T ( R ) } – in our setting this is given by T ℓ = M τ ∈ T ( ℓ,R ) T [ τ ] ,where T ( ℓ, R ) = { τ ∈ T ( R ) : deg( τ ) = ℓ } . In many applications the spaces T ℓ arefinite-dimensional and there is no need to specify the norm k • k ℓ on T ℓ (since theyare all equivalent). auge covariance While the spaces T ℓ are also finite-dimensional in our setting, we want to encodethe fact that K ε is converging to K and ξ ε is converging to ξ as ε ↓ in a way thatallows us to treat discrepancies between these quantities as small at the level of ourabstract formulation of the fixed point problem. We achieve this by defining, foreach ℓ ∈ deg( R ) , a family of norms {k • k ℓ,ε : ε ∈ ( , ] } on T ℓ . Our definition willdepend on a small parameter θ ∈ ( , κ ) which we treat as fixed in what follows.Heuristically and pretending for a moment that we are in the scalar noise setting,we define these k • k ℓ,ε norms by performing a “change of basis” and writing outtrees in terms of the noises Ξ i , ¯Ξ i − Ξ i , operators I i,p , ¯ I i,p − I i,p , I h i,p and I u instead of Ξ i , ¯Ξ i , I i,p , ¯ I i,p , I h i,p and I u , respectively. For instance, we rewrite ¯ I i ( ¯Ξ i ) = ( ¯ I i − I i )( ¯Ξ i − Ξ i ) + ( ¯ I i − I i ) Ξ i + I i ( ¯Ξ i − Ξ i ) + I i Ξ i . We then define, for any ℓ ∈ deg( R ) and v = P τ ∈ T ( ℓ,R ) v τ τ ∈ T ℓ , k v k ℓ,ε = max { ε m ( τ ) θ | v τ | : τ ∈ T ( ℓ, R ) } ,where m ( τ ) counts the number of occurrence of ¯ I i,p − I i,p and ¯Ξ i − Ξ i in τ .We now make this idea more precise and formulate it our setting of vector-valuednoise. Recall that in our new setting the trees serve as indices for subspaces of ourregularity structure, instead of basis vectors, so we do not really “change basis”.We note that there is a (unique) isomorphism Θ : T → T with the properties that • Θ preserves the domain of I i,p , I h i,p and I u and commutes with theseoperators on their domain. • For any τ, ¯ I i,j ( τ ) ∈ T ( R ) one has Θ ◦ ¯ I i,j = ( ¯ I i,j + I i,j ) ◦ Θ . • For any u, v ∈ T with uv ∈ T one has Θ ( u ) Θ ( v ) = Θ ( uv ) – here weare referencing the partially defined product on T induced by the partiallydefined tree product on T ( R ) . • The restriction of Θ to T [ ¯Ξ i ] is given by id + σ − where σ − is the inverseof the map σ given in (7.12). • Θ restricts to the corresponding identity map on both T [ X k ] and T [ Ξ i ] .It is immediate that Θ furthermore preserves T ℓ for every ℓ ∈ deg( R ) .We now fix, for every τ ∈ T ( R ) , some norm k • k τ on T [ τ ] . Since each T [ τ ] isisomorphic to a subspace of ( g ∗ ) ⊗ n and the isomorphism is furthermore canonicalup to permutation of the factors, this can be done by choosing a norm on g ∗ as wellas a choice of uniform crossnorm (for example the projective crossnorm).We then define a norm ⌊⌉ • ⌊⌉ ℓ,ε on T ℓ by setting, for any v ∈ T ℓ , ⌊⌉ v ⌊⌉ ℓ,ε = max { ε m ( τ ) θ k P τ v k τ : τ ∈ T ( R, ℓ ) } ,where P τ is the projection from T ℓ to T [ τ ] and now m ( τ ) counts the number ofoccurrences of the labels { ¯ l i , m i } di =1 appearing in τ . Finally, the norm k • k ℓ,ε isgiven by setting k v k ℓ,ε = ⌊⌉ Θ v ⌊⌉ ℓ,ε .The following lemma, which is straightforward to prove, states that these normshave the desired qualities. auge covariance • Let ℓ ∈ deg( R ) and v ∈ T ℓ with v is in the domain of the operator ¯ I i,p − I i,p .Then one has, uniform in ε , k ( ¯ I i,p − I i,p )( v ) k ℓ +2 − κ,ε . ε θ k v k ℓ,ε . (7.13) • For any u ∈ T [ Ξ i ] one has, uniformly in ε , k ( σ ( u ) − u ) k − d/ − − κ,ε . ε θ k u k − d/ − − κ,ε . (7.14)Once we fix these ε -dependent norms on our regularity structure we also obtaincorresponding • ε -dependent seminorms and metrics on models which we denote by k • k ε and d ε ( • , • ) respectively; and • ε -dependent norms on D γ,η ⋉ M ε which we denote by | • | γ,η,ε . Remark 7.5
Recall that modelled distributions in the scalar setting take values inthe regularity structure T and therefore in the definition of a norm on modelleddistributions we reference norms k • k ℓ on the spaces T ℓ . When we allow our noises /solutions to live in finite-dimensional vector spaces our modelled distributions willtake values in T ⊗ W for some finite-dimensional vector space W and so whenspecifying a norm on such modelled distributions we will need to reference normson T ℓ ⊗ W We assume that we have already fixed, for any such space W appearing, an ε -independent norm k • k W . Then we view our norm on T ℓ ⊗ W as induced bythe norm on T ℓ by taking some choice of crossnorm (the particular choice does notmatter). Remark 7.6
Clearly, all of our ε -dependent seminorms / metrics on models areequivalent for different values of ε ∈ ( , ] , but not uniformly so as ε ↓ . Thedistances for controlling models (resp. modelled distributions) become stronger(resp. weaker) as one takes ε smaller. Remark 7.7
In general, one would not expect the estimates of the extension the-orem [Hai14, Theorem 5.14] to hold uniformly as we take ε ↓ . However, it isstraightforward to see from the proof of [Hai14, Theorem 5.14] that they do holduniformly in ε for models in M ε (and, more trivially, M ) thanks to Lemma 7.3and the fact that θ ≤ κ . For sufficiently small ˜ θ > , one has a classical Schauder estimate | G ∗ f − G ε ∗ f | C ℓ . ε ˜ θ | f | C ℓ +˜ θ , auge covariance which holds for all distributions f and non-integer regularity exponents. Our con-ditions on our ε -dependent norms let us prove an analogous estimate at the level ofmodelled distributions. In what follows we write K i , ¯ K i for the abstract integrationoperators on modelled distributions associated to a i , and m i , respectively. Lemma 7.8
Fix i ∈ [ d ] and let V be a sector of regularity α in our regularitystructure which is in the domain of both I i and ¯ I i . Fix γ ∈ ( , ] and η < γ suchthat γ + 2 − κ N , η + 2 − κ N , and η ∧ α > − .Then, for fixed M > , one has | K i H − ¯ K i H | γ +2 − κ, ¯ η,ε . ε θ | H | γ,η,ε uniformly in ε ∈ ( , ] , Z ∈ M ε with k Z k ε ≤ M , and H ∈ D γ,η ( V ) ⋉ Z , where θ ∈ ( , κ ) is the fixed small parameter as above and ¯ η = ( η ∧ α ) + 2 − κ .Proof. This result follows from the proof of [Hai14, Theorem 5.12 and Proposi-tion 6.16]. Indeed, in the context of this reference, and working with some fixednorm on the given regularity structure, if the abstract integrator I ( · ) of order β inquestion has an operator norm (as an operator on the regularity structure) boundedby ˜ M , and the kernel I realises is ( γ + β, ˜ M , β ) -regularising, then as long as γ + β N and η + β N , one has | K H | γ + β, ( η ∧ α ) + β . ˜ M | H | γ,η . Here, K is the corresponding integration on modelled distributions and the propor-tionality constant only depends on the size of the model in the model norm (whichcorresponds to the fixed norm on the regularity structure).Our result then follows by combining this observation with the fact that we canview I i − ¯ I i as an abstract integrator of order − κ on our regularity structurerealising the kernel K − K ε which is ( m, ε κ R, − κ ) -regularising by Lemma 7.3,and the fact that I i − ¯ I i has norm bounded by ε θ by (7.13).We specialise the above lemma to the particular estimate that we will need in ourcomparison of abstract fixed point problems. We first define, as in Section 6, Ξ i ∈ T [ Ξ i ] ⊗ g and ¯Ξ i ∈ T [ ¯Ξ i ] ⊗ g to be given by “ id g ” via the canonicalisomorphisms T [ ¯Ξ i ] ⊗ g ≃ T [ Ξ i ] ⊗ g ≃ g ∗ ⊗ g ≃ L ( g , g ) . Note that wehave σ Ξ i = ¯Ξ i where we continue our abuse of notation with σ acting onlyon the left factor. We also remark that, as g -valued modelled distributions, Ξ i , ¯Ξ i ∈ D ∞ , ∞− d/ − − κ . Then, thanks to (7.14) and Remark 7.5 one has, uniform in ε ∈ ( , ] , k Ξ i − ¯Ξ i k ∞ , ∞ ,ε . ε θ . (7.15) auge covariance Fix i ∈ [ d ] . Let γ be such that γ − d − − κ ∈ ( , ) and η > . Let ¯ γ = γ − d + 1 − κ / ∈ N and ¯ η = η − d + 1 − κ / ∈ N . Then, for any M > , onehas | K i ( H ¯Ξ i ) − ¯ K i ( H Ξ i ) | ¯ γ, ¯ η,ε . ε θ | H | γ,η,ε , (7.16) uniformly over all ε ∈ ( , ] , Z ∈ M ε with k Z k ε ≤ M , and L ( g , g ) -valuedmodelled distributions H ∈ D γ,η ( V ) ⋉ Z . Here V is a sector of regularity which admits multiplication with any element of T [ Ξ i ] or T [ ¯Ξ i ] and subsequentintegration with I i or ¯ I i .Proof. We write K i ( H ¯Ξ i ) − ¯ K i ( H Ξ i ) = K i ( H ( ¯Ξ i − Ξ i )) + ( K i − ¯ K i )( H Ξ i ) . (7.17)Note that | K i ( H ( Ξ i − ¯Ξ i )) | ¯ γ, ¯ η,ε . | H ( Ξ i − ¯Ξ i ) | γ − d − − κ, η − d − − κ, ε . | H | γ,η,ε | Ξ i − ¯Ξ i | ∞ , ∞ ,ε . ε θ | H | γ,η,ε ,where we used the standard multi-level Schauder estimate [Hai14, Proposition 6.16]in the first inequality, the standard multiplication bounds [Hai14, Proposition 6.12]in the second inequality, and (7.15).To finish the proof we observe that | ( K i − ¯ K i )( H Ξ i ) | ¯ γ, ¯ η,ε . ε θ | H Ξ i | γ − d − − κ, η − d − − κ, ε . ε θ | H | γ,η,ε ,where we used Lemma 7.8 in the first inequality and the standard multiplicationbound in the second inequality.We now write out the analytic fixed point problems for (7.2) and (7.3). We in-troduced the labels m i just to assist with deriving the renormalised equation andso when we pose our analytic fixed point problem we stray from the formulationgiven in Remark 5.64 and instead eliminate the components m i appearing in (7.3)by performing a substitution.In what follows, we write R for the reconstruction operator. Recall that K i , ¯ K i are the abstract integration operators associated to a i and m i ; we also write K h i and K u for the abstract integration operators on modelled distributions correspondingto I h i and I u , and R the operator realising convolution with G − K as a map fromappropriate HÃűlder–Besov functions into modelled distributions as in [Hai14,(7.7)].Given initial data ( B ( ) , U ( ) , h ( ) ) ∈ Ω C η × C α ( T , L ( g , g )) × C α − ( T , g ) , (7.18)the fixed point problem associated with (7.2) for the g -valued modelled distributions ( B i ) di =1 , ( H i ) di =1 and L ( g , g ) -valued modelled distribution U is B i = G i + (cid:16) [ B j , ∂ j B i − ∂ i B j + [ B j , B i ]] auge covariance + ˚ C B i + ˚ C H i + U ¯Ξ i (cid:17) + GB ( ) i (7.19) H i = G h i + (cid:16) [ H j , ∂ j H i ] + [[ B j , H j ] , H i ] + ∂ i [ B j , H j ] (cid:17) + Gh ( ) i U = G u + (cid:16) − [ H j , [ H j , · ]] ◦ U + [[ B j , H j ] , · ] ◦ U (cid:17) + GU ( ) ,where + is the map that restricts modelled distributions to non-negative times, G t def = K t + R R and finally G • refers to the “harmonic extension” map of [Hai14,(7.13)].The modelled distribution fixed point problem for the ( ¯ A, ¯ U , ¯ h ) system (7.3) isthe same as (7.19) except that the first equation is replaced by B i = G i + (cid:16) [ B j , ∂ j B i − ∂ i B j + [ B j , B i ]] + ˚ C B i + ˚ C H i (cid:17) (7.20) + ¯ G i + ( U Ξ i ) + GB ( ) i ,where ¯ G i def = ¯ K i + ¯ R R with ¯ R defined just like R but with G − K replaced by G ε − K ε . In (7.20) we have written B i instead of something like ¯ A i to make itclearer that we are comparing two fixed point problems which have “almost” thesame form – only the terms G i + (cid:0) U ¯Ξ i (cid:1) and ¯ G i + ( U Ξ i ) are different. We can nowmake precise what we mean by the two problems being “close”. Lemma 7.10
For initial data ( B ( ) , U ( ) , h ( ) ) as in (7.18) , the fixed point problems (7.19) and (7.20) are well-posed on the bundle of modelled distributions (cid:16) M t = a i , h i , u D γ t ,η t α t (cid:17) ⋉ M where ( γ t , α t , η t ) = ( κ, − κ, η ) if t = a i , ( κ, , η + 1 ) if t = u , ( κ, , η ) if t = h i (7.21) For any T ∈ ( , ] and L > , let S L,T ( • ) and ¯ S L,T ( • ) be the correspondingsolution maps with cut-off size L and cut-off time T for systems (7.19) and (7.20) .Then, for any ¯ R > , and uniform in ε ∈ ( , ] with Z ∈ M ε with k Z k ε ≤ ¯ R we have the estimate |S L,T ( Z ) − ¯ S L,T ( Z ) | ~γ,~η,ε . ε θ and kR Z S L ( Z ) − R Z ¯ S L ( Z ) k ~α . ε θ , (7.22) where | • | ~γ,~η,ε is a corresponding multi-component modelled distribution norm for (7.21) , R Z is the reconstruction operator associated to Z , and k • k ~α is the normon M t = a i , h i , u C α t ( [ , T ] × T d , W t ) . Proof.
We first note that the two fixed point problems differ in the B i components,namely by the two terms K i + ( U ¯Ξ i ) − ¯ K i + ( U Ξ i ) and R R + (cid:0) U ¯Ξ i (cid:1) − ¯ R R + ( U Ξ i ) . auge covariance By Lemma 7.9, the first term gives a contribution of order ε θ . The second termfollows by an analogous argument as the one used in the proof of Lemma 7.9 buteasier since we are dealing with the smooth part of the integration map. Namelywe first write the second term as R R + U ( Ξ i − ¯Ξ i ) + ( R − ¯ R ) R + U Ξ i . Thefirst piece can be estimated in the same way as before but using the estimate of[Hai14, Lemma 7.3] instead of the Schauder estimate for K i . It is straightforward,by referring to the definition [Hai14, (7.7)] of R − ¯ R , to argue that the second pieceis of order ε since, for any α > , one has k ( G − K ) − ( G ε − K ε ) k C α . ε .With these bounds in hand, the first estimate of (7.22) then follows by thestability of the fixed point established in the proof of [Hai14, Theorem 7.8] andthe second estimate is a consequence of the reconstruction bound (note that thereconstruction bound is uniform in ε even though we are using ε -dependent normson both models and modelled distributions). Our key input in our argument regarding stochastic control of our models is givenby the following lemma.
Lemma 7.11
One has, for any p ≥ , sup ε ∈ ( , ] sup δ ∈ ( ,ε ) E [ k Z δ,ε bphz k pε ] < ∞ . (7.23) Moreover, there exist models Z ,ε bphz ∈ M ε for ε ∈ ( , ] such that, for any such ε , lim δ ↓ Z δ,ε bphz = Z ,ε bphz (7.24) in probability with respect to the topology of d ε ( • , • ) .Finally, there exists a model Z , bphz ∈ M such that lim ε ↓ Z ,ε bphz = Z , bphz (7.25) in probability with respect to the topology of d ( • , • ) .Proof. As in Lemma 6.4 we proceed by using the results of [CH16]. We start byproving (7.24) and here we appeal to [CH16, Theorem 2.15]. We first note thatfor any scalar noise decomposition, it is straightforward to verify that the randomsmooth noise assignments ζ δ,ε are a uniformly compatible family of Gaussian noisesthat converge to the Gaussian noise ζ ,ε . The verification of the first three listedpower-counting conditions of [CH16, Theorem 2.15] is analogous to how they werechecked for Lemma 6.4. This gives the existence of the limiting models Z ,ε bphz andthe desired convergence statement (note that for fixed ε > , the metric d ε ( • , • ) isequivalent to d ( • , • ) ). auge covariance To prove (7.25) we will show lim ε ↓ sup δ, ˜ ε ∈ ( ,ε ) E [ d ( Z δ,ε bphz , Z δ, ˜ ε bphz ) ] = 0 . (7.26)By using Fatou to take the limit δ ↓ , this gives us that Z ,ε is Cauchy in L as ε ↓ and so we obtain the desired limiting model Z , bphz and the desired convergencestatement.To prove (7.26) we will use the more quantitative [CH16, Theorem 2.31]. Herewe take L cum to be the set of all pairings of L − and so the three power-countingconditions we verified for [CH16, Theorem 2.15] also imply the super-regularityassumption of [CH16, Theorem 2.31]. Since we only work with pairings, thecumulant homogeneity c is determined by our degree assignment on our noises.After rewriting the difference of the action of models as a telescoping sum whichallows one to factor the corresponding difference in the kernel assignment K − K ε or noise assignment ξ ˜ εi − ξ εi , one is guaranteed at least one factor of order ε κ theright-hand side of the bound [CH16, (2.15)] – coming from k K − K ε k − κ,k in thefirst case or the contraction ξ ˜ εi − ξ εi with another noise measured in the k • k − − κ,k kernel norm in the second case. This gives us the estimate (7.26).The above argument for obtaining (7.26) can also be applied to obtain (7.23),namely, with the constraint that δ ∈ ( , ε ) , occurrence of I t ,p − ¯ I t ,p gives a factorof ε κ through the difference K − K ε and any occurrence of Ξ l i − Ξ ¯ l i gives a factor ε κ through the difference ξ δi − ξ εi and since θ ∈ ( , κ ) this gives the suitable uniformin ε bounds on the moments of the model norm k • k ε . In this section we derive the renormalised equations for the B system and the ¯ A system and prove that they converge to the same limit, i.e. Proposition 7.22. Given δ ∈ [ , ] and ε ∈ ( , ] , we write ℓ δ,ε bphz [ • ] for the BPHZ renormalisationgroup character that goes between the canonical lift and Z δ,ε bphz . The rule givenbelow (7.9) determines the set T − ( R ) of trees as in (5.22) and we only list the treesin T − ( R ) that are relevant to deriving the renormalised equations in the followingtwo tables (for the F system and the ¯ F system respectively). The reason that wewill only need to be concerned with these trees will be clear by Lemma 7.12 below,which follows easily from the definition of Υ • t [ • ] and the parity constraints on thenoises and spatial derivatives that are necessary for ℓ δ,ε bphz [ • ] not to vanish.Here the graphic notation is similarly as in Section 6: (thick) lines denote(derivatives of) I , colors denote spatial indices, and the color of a tiny trianglelabels the spatial index for the kernel immediately below it. Moreover, we drawa circle (resp. crossed circle) for ¯Ξ (resp. X ¯Ξ ), with a convention that the lineimmediately below it understood as I , and a square (resp. crossed square) for Ξ (resp. X Ξ ), with a convention that the line immediately below it understood as ¯ I . auge covariance We also draw a zigzag line for I u and a wavy line for I h . Their thick versionsand tiny triangles above them are understood as before.Table 1 I ( ¯Ξ ) I ( ¯Ξ ) I , ( I , ( ¯Ξ )) I , ( ¯Ξ ) I ( I , ( ¯Ξ )) I ( ¯Ξ ) I , ( X ¯Ξ ) I ( X ¯Ξ ) I , ( ¯Ξ ) I u ( I ( ¯Ξ )) ¯Ξ I , ( ¯Ξ ) I h ( I , ( ¯Ξ )) I ( ¯Ξ ) I h , ( I , ( ¯Ξ )) Table 2 ¯ I ( Ξ ) ¯ I ( Ξ ) I , ( ¯ I , ( Ξ )) ¯ I , ( Ξ ) I ( ¯ I , ( Ξ )) ¯ I ( Ξ ) ¯ I , ( X Ξ ) ¯ I ( X Ξ ) ¯ I , ( Ξ ) I u ( ¯ I ( Ξ )) Ξ¯ I , ( Ξ ) I h ( ¯ I , ( Ξ )) ¯ I ( Ξ ) I h , ( ¯ I , ( Ξ )) The first five trees in each of the two tables have the same structure as the onesthat appeared in Section 6, except that now the noises are understood as Ξ or ¯Ξ , andedges understood as I or ¯ I . An important difference from Section 6 is that thetrees of the type and had vanishing Υ in Section 6 and therefore no effect onthe renormalised equation, but this is not the case now, as we will see below, due tothe term U J ε ξ (or J ε ( ¯ U ξ ) ) in our equation. Moreover, the tables also show trees in T − ( R ) such as those of the form which do not have any counterpart in Section 6. Lemma 7.12 If τ ∈ T − ( R ) is not of any of the forms listed in Table 1 (resp. Table 2)then either ℓ δ,ε bphz [ τ ] = 0 or Υ F t [ τ ] = 0 (resp. either ℓ δ,ε bphz [ τ ] = 0 or Υ ¯ F t [ τ ] = 0 )for every t ∈ L + .Proof. The proof of this lemma follows similar lines as Lemma 6.5, so we do norepeat the details. We only remark that for trees with a “polynomial” X , namely I , ( ¯Ξ ) I ( X ¯Ξ ) , I ( ¯Ξ ) I , ( X ¯Ξ ) , ¯ I , ( Ξ ) ¯ I ( X Ξ ) , ¯ I ( Ξ ) ¯ I , ( X Ξ ) , the polynomial can be dealt with in the same way as for the derivative in Lemma 6.5;for instance for the first tree, if = , then flipping the sign of the -component (or,-component) of the appropriate integration variable shows that ¯ Π can [ τ ] = 0 .We now state a sequence of lemmas with identities for ¯Υ F and ¯Υ ¯ F , but wewill not give the detailed calculations within the proof of each lemma, since theseare straightforward (for instance they follow similarly as in Section 6). We firstshow that in both F and ¯ F systems we don’t see any renormalisation of the u or h i equations. auge covariance For any of the τ of the form listed in Table 1 (resp. Table 2) one has ¯ Υ F u [ τ ] = 0 (resp. ¯ Υ ¯ F u [ τ ] = 0 ). Moreover, we have X τ ∈ T − ( R ) ( ℓ δ,ε bphz [ τ ] ⊗ id) ¯ Υ F h , [ τ ]( A ) = X τ ∈ T − ( R ) ( ℓ δ,ε bphz [ τ ] ⊗ id) ¯ Υ ¯ F h , [ τ ]( A ) = 0 . Proof.
The fact that ¯ Υ F u [ τ ] = ¯ Υ ¯ F u [ τ ] = 0 for τ appearing in the tables followsfrom direct computation. One has ¯ Υ F h , [ τ ] = 0 (resp. ¯ Υ ¯ F h , [ τ ] = 0 ) for any τ in Table 1 (resp. Table 2) of the first six shapes. For the other trees one has, byintegration by parts, ℓ δ,ε bphz [ ] = − ℓ δ,ε bphz [ ] , ℓ δ,ε bphz [ ] = − ℓ δ,ε bphz [ ] . (7.27)Additionally, one has Υ h , [ ] = Υ h , [ ] , Υ h , [ ] = Υ h , [ ] . (7.28)Above we are exploiting the canonical isomorphisms between the spaces where theobjects above live – namely for any two trees τ , ¯ τ of any of the four forms appearingabove, one has a canonical isomorphism T [ τ ] ≃ T [ ¯ τ ] by using Remark 5.11 andthe canonical isomorphisms between these trees obtained by only keeping their treestructure. Combining (7.27) with (7.28) then yields the last claim.We define a subset ¯ A ⊂ A that encodes additional constraints on the jet of oursolutions which come from (7.1) and (7.5). These constraints will also help ussimplify the counterterms for the a i and m i equations. Definition 7.14
We define ¯ A to be the collection of all A = ( A o ) o ∈E ∈ A such that • A u is unitary. • For all a, b ∈ g , A u [ a, b ] = [ A u a, A u b ] . • A ∂ j u = [ A h j , · ] ◦ A u . Remark 7.15
Lemma 7.13 guarantees that the renormalised reconstruction ob-tained via the models ( Z ,ε bphz : ε ∈ [ , )) , of our equations for U and h (resp. ¯ U and ¯ h ) will be the same as what appears for these components in (7.2).This observation means that, for initial data for U and h satisfying (7.1) for somefixed initial g ( ) , the abstract solution for the F system obtained via the models ( Z ,ε bphz : ε ∈ [ , ]) , together with their derivatives, will take values in ¯ A pointwise.The analogous statement holds true for ¯ U and ¯ h equations and the ¯ F system.To argue this we first note that, since the constraint imposed by ¯ A defines aclosed set and the abstract solution map is continuous with respect to the model, itsuffices to prove the claim when ε > . In this case, irrespective of the form ofthe renormalised equation for B , we can repeat the computations of Lemma 7.1 toshow that if we start with initial U ( ) and h ( ) as above then, for positive existence auge covariance times t > , U ( t ) and h ( t ) satisfy (7.1) with respect to the gauge transformation g ( t ) given by the evolution of g ( ) by the g equation given in (2.5) for some smoothprocess B .This means that, for the sake of proving Theorem 2.8, we can assume therelations given in Definition 7.14 hold when computing the renormalised equationsfor B and ¯ A for the models ( Z ,ε bphz : ε ∈ [ , )) .We now turn to explicitly identifying the renormalisation counterterms for the a i and m i equations in the ¯ F system.We start by collecting formulae for the the renormalisation constants. Write K δ,ε = K ε ∗ χ δ = K ∗ χ ε ∗ χ δ and recall the constants ˆ C ε and ¯ C ε defined in (6.12).We then define the variants ¯ C δ,ε def = Z dz K δ,ε ( z ) , ˆ C δ,ε def = Z dz ∂ j K δ,ε ( z )( ∂ j K ∗ K δ,ε )( z ) , (7.29)where one can choose any j ∈ { , } as in (6.12). We then have the followinglemma. Lemma 7.16
For ˆ C ε and ¯ C ε as in (6.12) , one has ℓ δ,ε bphz [ ] = − ℓ δ,ε bphz [ ] = ˆ C ε Cas , ℓ δ,ε bphz [ ] = ¯ C ε Cas . (7.30) For ˆ C δ,ε and ¯ C δ,ε defined as in (7.29) one has ℓ δ,ε bphz [ ] = − ℓ δ,ε bphz [ ] = ˆ C δ,ε Cas , ℓ δ,ε bphz [ ] = ¯ C δ,ε Cas . (7.31) Finally, for any ε > , one has lim δ ↓ ˆ C δ,ε = ˆ C ε and lim δ ↓ ¯ C δ,ε = ¯ C ε .Proof. The statements (7.30) and (7.31) follow in the same way as Lemma 6.9.The final statement about convergence as δ ↓ of the renormalisation constants isobvious.We introduce additional renormalisation constants ˜ C ε def = Z dz χ ε ( z )( K ∗ K ε )( z ) , ˜ C δ,ε def = Z dz χ δ ( z )( K ∗ K δ,ε )( z ) . The following lemma is straightforward to prove.
Lemma 7.17
One has ℓ δ,ε bphz [ ] = ˜ C ε Cas , ℓ δ,ε bphz [ ] = ˜ C δ,ε Cas ,and furthermore lim δ ↓ ˜ C δ,ε = ( K ∗ K ε )( ) def = ˜ C ,ε . Additionally, there are finiteconstants C gsym and ¯ C gsym such that lim ε ↓ ˜ C ε = C gsym and lim ε ↓ ˜ C ,ε = ¯ C gsym . auge covariance Finally, we have that ℓ δ,ε bphz [ ] , ℓ δ,ε bphz [ ] , ℓ δ,ε bphz [ ] , and ℓ δ,ε bphz [ ] are eachgiven by a multiple of Cas where the prefactor only depends on δ, ε and the form of the tree. The rest of our computation of the renormalised equation is summarised in thefollowing lemmas. In what follows we refer to the constant λ fixed by Remark 6.8.We also introduce the shorthand Ψ = I ¯ Ξ , Ψ , = I , ¯ Ξ , ¯Ψ = ¯ I Ξ , ¯Ψ , = ¯ I , Ξ . We now walk through the computation of renormalisation counterterms for thesystem of equations given by ¯ F . We will directly give the expressions for ¯ Υ ¯ F suchas (7.32) and (7.35) below, which follow by straightforward calculations from thedefinitions.Recall the convention (7.10) for writing components of A as B , ¯ A , U , etc. Thefollowing lemma gives the renormalisation for the m i equation in this system. Lemma 7.18 ¯ Υ ¯ F m , [ τ ] = 0 for all τ of the form in Table 2 except for τ = where ¯ Υ ¯ F m , [ ]( A ) = δ , [ [ ¯ U I u ¯Ψ , ¯ h ] , ¯ U Ξ ] . (7.32) In particular, for A ∈ ¯ A , X τ ∈ T − ( R ) ( ℓ δ,ε bphz [ τ ] ⊗ id) ¯ Υ ¯ F m , [ τ ]( A ) = − λ ˜ C δ,ε ¯ h . (7.33) Proof.
Using the assumption that A ∈ ¯ A we have [ [ ¯ U I u ( ¯Ψ ) , ¯ h ] , ¯ U Ξ ] = − ¯ U [ Ξ , [ I u ( ¯Ψ ) , ¯ U − ¯ h ] ] . Inserting this into the left-hand side of (7.33) and combining it with Lemma 7.17,we see that it is equal to − ˜ C δ,ε (Cas ⊗ id) ¯ U [ Ξ , [ I u ( ¯Ψ ) , ¯ U − ¯ h ]] = − ˜ C δ,ε ¯ U ad Cas ¯ U − ¯ h = − λ ˜ C δ,ε ¯ h since ad Cas = λ id g .For the a i components we have the following lemma. Lemma 7.19 ¯ Υ ¯ F a , [ ] = 0 , and ¯ Υ ¯ F a , [ ]( A ) = = [ ¯ U ¯Ψ , [ ¯ U ¯Ψ , ¯ A ]] , (7.34) That is, they do not depend on the specific colors / spatial indices appearing in the tree as long asthey obey the constraints given in Tables 1 and 2. Note that our use of the notations Ψ and Ψ , differs slightly from Section 6. auge covariance ¯ Υ ¯ F a , [ ]( A ) = ( δ , δ , − δ , δ , ) [[ δ , ¯ A − δ , ¯ A , ¯ U I ( ¯Ψ , )] , ¯ U ¯Ψ , ] , ¯ Υ ¯ F a , [ ]( A ) = ( δ , δ , − δ , δ , )[ ¯ U ¯Ψ , [ δ , ¯ A − δ , ¯ A , ¯ U I , ( ¯Ψ , )]] , ¯ Υ ¯ F a , [ ]( A ) = δ , ( δ , − )[ ¯ U ¯Ψ , ∂ ¯ U ¯ I , ( X Ξ )] , (7.35) ¯ Υ ¯ F a , [ ]( A ) = δ , ( δ , − )[ ∂ ¯ U ¯ I ( X Ξ ) , ¯ U ¯Ψ , ] . (7.36) In particular, for A ∈ ¯ A , X τ ∈ T − ( R ) ( ℓ δ,ε bphz [ τ ] ⊗ id) ¯ Υ ¯ F a , [ τ ]( A ) = ( ¯ C δ,ε − C δ,ε ) λ ¯ A . (7.37)
Proof.
The right-hand side of (7.37) comes from the contribution of trees of theform , , and , which can be shown as in Lemma 6.14, combined with thecondition that A ∈ ¯ A (namely, the second relation of Definition 7.14) to cancel thefactors of ¯ U . The total contributions from the trees of the form and those ofthe form each vanish. For the case of trees of form this total contribution isgiven by ˇ C X =1 , ( δ , − )(Cas ⊗ id)[ ¯ U ¯Ψ , ∂ ¯ U ¯ I , ( X Ξ )] ,for some constant ˇ C . Other than the factor ( δ , − ) , the summand above does notdepend on and since P =1 , ( δ , − ) = 0 it follows that the sum above vanishesas claimed. A similar argument takes care of the case of .The computation of the renormalisation of the a i components in the F system ofequations mirrors the computations we have just done for the ¯ F system with theone difference that the term U ¯ ξ i , which is the analogue of the term U ξ i that waspart of ¯ F m ,i , is included in F a ,i . In particular, ¯ Υ F a [ τ ] for τ ∈ { , , , , } are given by formulas as in (7.34) and (7.35) with the following replacement ¯ A B, ¯ U U , ¯Ψ Ψ , ¯ I ( X Ξ ) I ( X ¯ Ξ ) and ¯ Υ F a , [ ]( A ) = δ , [[ U I u Ψ , h ] , U ¯ Ξ ] .By using the renormalisation constants given in Lemma 7.17 and performingagain computations of the type found in Lemmas 7.18 and 7.19, one obtains thefollowing lemma. Lemma 7.20
For A ∈ ¯ A , X τ ∈ T − ( R ) ( ℓ δ,ε bphz [ τ ] ⊗ id) ¯ Υ F a , [ τ ]( A ) = λC ε sym B − λ ˜ C ε h ,where C ε sym is as in (6.12) . Remark 7.21
Lemmas 7.18, 7.19, and 7.20, still hold if one replaces the firstcondition of Definition 7.14 by only requiring the invertibility of A u . auge covariance The main result of this section is the following proposition.
Proposition 7.22
Fix any constants ˚ C and ˚ C and initial data ¯ a ∈ Ω α and g ( ) ∈ G α . Consider the system of equations ∂ t ¯ A i = ∆ ¯ A i + χ ε ∗ ( ¯ gξ i ¯ g − ) + ¯ C ε ¯ A i + ¯ C ε ( ∂ i ¯ g ) ¯ g − (7.38) + [ ¯ A j , ∂ j ¯ A i − ∂ i ¯ A j + [ ¯ A j , ¯ A i ]] , ¯ A ( ) = ¯ a ,and ∂ t B i = ∆ B i + gξ εi g − + C ε B i + C ε ( ∂ i g ) g − (7.39) + [ B j , ∂ j B i − ∂ i B j + [ B j , B i ]] , B ( ) = ¯ a ,where ¯ g and g are given by running the corresponding equations in (2.6) and (2.5) started with the same initial data ¯ g ( ) = g ( ) and we have defined the constants ¯ C ε = ˚ C + λC ε sym , ¯ C ε = ˚ C − λ ˜ C ,ε , (7.40) C ε = ˚ C + λC ε sym , C ε = ˚ C − λ ˜ C ε . Then, ¯ A and B converge in probability in Ω sol to the same limit as ε ↓ .Proof. We claim that (7.38) is just the renormalised equation obtained via thereconstruction (with respect to Z ,ε bphz ) of the fixed point problem (7.20). Since Z ,ε bphz is not a smooth model, the justification of this claim goes via obtaining thecorresponding renormalised equation for the model Z δ,ε bphz and then taking the limit δ ↓ (which is justified by the convergence (7.24)).We deploy [BCCH17, Thm. 5.7] and Proposition 5.65 to get the renormalisedreconstruction of the equation (7.20). In terms of the indeterminates A = ( A o ) o ∈E and nonlinearity ¯ F , this amounts to summing the renormalised and reconstructedintegral fixed point equations for the indeterminates A a i and A m i with nonlinearity ¯ F , and recalling (7.11).The claim then follows by using Lemma 7.13 and Remark 7.15 to allow us togo between Ad ¯ g and ( ∂ i ¯ g ) ¯ g − and ¯ U and ¯ h i , then using the explicit computationsof counter-terms in Lemmas 7.18 and 7.19, and then taking the limit δ ↓ ofrenormalisation constants as given in Lemmas 7.16 and 7.17.A similar argument shows that (7.39) is the renormalised equation obtained viathe reconstruction (with respect to Z ,ε bphz ) of the fixed point problem (7.19) withthe minor differences that one is aiming for the renormalised and reconstructedintegral fixed point equations for just the indeterminates A a i with non-linearity F so the computations of Lemma 7.18 and 7.19 are replaced by that of Lemma 7.20.We now turn to proving the statements concerning convergence in probabilityas ε ↓ . We first show that the statement holds if, in the definition of Ω sol ,one replaced Ω α,T with C ([ , T ) , C α − ) . The convergence of ¯ A and B individuallyfollow from the convergence of the models Z ,ε bphz given in Lemma 7.11 and standardarguments using the continuity of the machinery of regularity structures as given in auge covariance [Hai14]. The statement that d ( ¯ A, B ) → in probability as ε ↓ follows from thesecond estimate of (7.22) from the statement of Lemma 7.9. To obtain the controlover models needed to apply this lemma it suffices to point out that by combiningstatements (7.23) and (7.24) of Lemma 7.11 we have, for any p ≥ , sup ε ∈ ( , ] E [ k Z ,ε bphz k pε ] < ∞ . (7.41)To prove the desired statement for Ω sol we first note that by using the same argumentused in the proof of Theorem 2.4 at the end of Section 6.2.5 (namely, splitting intoa linear part with more regular remainder) one can show ¯ A, B both individuallyconverge in probability in Ω sol as ε ↓ .To show that d ( ¯ A, B ) → in probability as ε ↓ we first fix α ′ ∈ ( , α ) andnote that every ball in Ω α is compact in Ω α ′ , and thus also in C α ′ − . Since anytwo comparable Hausdorff topologies on a set which render it compact coincide,convergence in C α ′ − with uniform bounds in Ω α implies convergence in Ω α ′ .Hence, since | ¯ A ( t ) − B ( t ) | C α ′− → and B ( t ) and ¯ A ( t ) a.s. stay bounded in a ballin Ω α for each t ∈ [ , T ∗ ) as ε → , we obtain | ¯ A ( t ) − B ( t ) | α ′ → . Proof of Theorem 2.8.
We first prove statement (i). Given C ∈ R , which is assumedto be a real constant by Remark 2.7 and Assumption 6.1, we fix ˚ C = C − λC sym and ˚ C = C + λC gsym . We then take the ¯ C as claimed in the theorem as ¯ C def = λ ( ¯ C gsym − C gsym ) . With these choices and the definitions of (7.40), together with ˜ C ε − C gsym = o ( ) , C sym − C ε sym = o ( ) ,it follows that, as ε ↓ , C = C ε + o ( ) = C ε + o ( ) = ¯ C ε + o ( ) , C − ¯ C = ¯ C ε + o ( ) . The desired statement then follows from Proposition 7.22.We now prove (ii). Note that if χ is non-anticipative, then ˜ C ,ε = 0 for every ε > and so ¯ C gsym = 0 . It follows that ¯ C = − λC gsym = − λ lim ε ↓ R dz χ ε ( z )( K ∗ K ε )( z ) and so the desired statement follows from Remark 6.16. In this subsection, we prove Theorem 2.12. We begin with several lemmas.
Lemma 7.23
Let α ∈ ( , ] and A, B ∈ Ω α . Then (cid:12)(cid:12)(cid:12) inf g ∈ G ,α | B g | α − inf g ∈ G ,α | A g | α (cid:12)(cid:12)(cid:12) . ( | A | α + | B | α ) | A − B | α , (7.42) where the proportionality constant depends only on α . auge covariance Proof.
As in the proof of Theorem 3.27, for g ∈ G ,α we can write A g − B g = ( ( A − B ) g − g − ( A − B ) ) + ( A − B ) ,from which it follows by Lemmas 3.32 and 3.33 that | A g − B g | α . ( | g | α -Höl ) | A − B | α , (7.43)where the proportionality constant depends only on α . Consider a minimisingsequence g n ∈ G ,α for A . Then, by Proposition 3.35, limsup n →∞ | g n | α -Höl . | A | α , and thus by (7.43) inf g ∈ G ,α | B g | α − inf g ∈ G ,α | A g | α ≤ limsup n →∞ | B g n | α − | A g n | α . ( | A | α ) | A − B | α . Swapping A and B and applying the same argument, we obtain (7.42). Lemma 7.24
Let λ > . Then there exists a measurable (Borel) selection S : O α → Ω α such that | S ( x ) | α ≤ λ inf A ∈ x | A | α for all x ∈ O α .Proof. Consider the subset Y def = { A ∈ Ω α | | A | α ≤ λ inf g ∈ G ,α | A g | α } , which isclosed due to Lemma 7.23. In particular, Y is Polish and, by Lemma 3.40, thegauge equivalence classes in Y are closed. Finally, since π − ( π ( U )) = ∪ g ∈ G ,α U g is open for every open subset U ⊂ Ω α , the conclusion follows by the Rokhlin–Kuratowski–Ryll-Nardzewski selection theorem [Bog07, Thm. 6.9.3].For the rest of the section, let us fix a non-anticipative mollifier χ and set C = ¯ C ,the constant from part (i) of Theorem 2.8. By a “white noise” we again mean a pairof i.i.d. g -valued white noises ξ = ( ξ , ξ ) on R × T . Proof of Theorem 2.12. (i) By Lemma 7.24, there exists a measurable selection S : ˆ O α → ˆΩ α such that for all x ∈ O α | S ( x ) | α ≤ inf a ∈ x | a | α (7.44)and S ( ) = . Let ξ be white noise and let ( F t ) t ≥ be the filtration generated by ξ .Consider any a ∈ ˆΩ α . We define a càdlàg Markov process A : R + → ˆΩ α and a sequence of stopping times ( σ j ) ∞ j =0 as follows. For j = 0 , set σ = 0 and A ( ) = a . Consider now j ≥ . If σ j = ∞ , then we set σ j +1 = ∞ . Otherwise, if σ j < ∞ , suppose that A is defined on [ , σ j ] . If A ( σ j ) = , then define σ j = σ j +1 .Otherwise, define Θ ∈ C ([ σ j , ∞ ) , ˆΩ α ) by Θ ( t ) = Φ σ j ,t ( A ( σ j )) , where we used thenotation Φ s,t as in Definition 2.10, and set σ j +1 = inf { t > σ j | | Θ ( t ) | α > inf g ∈ G ,α | Θ ( t ) g | α } . We then define A ( t ) = Θ ( t ) for all t ∈ ( σ j , σ j +1 ) and A ( σ j +1 ) = S ([ Θ ( σ j +1 )]) .Observe that ( σ j , σ j +1 ) is a.s. non-empty due to Lemma 7.23, the condition (7.44), auge covariance and the continuity of Θ at σ j . In fact, defining M ( t ) = inf g ∈ G ,α | A ( t ) g | α , thenby decomposing Θ into the SHE with initial condition A ( σ j ) and a remainder asin the proof of Theorem 2.4, we see that the law of σ j +1 − σ j depends only on A ( σ j ) and can be stochastically bounded from below by a strictly positive randomvariable depending only on M ( σ j ) . In particular, if the quantity T ∗ def = lim j →∞ σ j is finite, then a.s. lim t ր T ∗ M ( t ) = ∞ . In this case, we have defined A on [ , T ∗ ) ,and then set A ≡ on [ T ∗ , ∞ ) . If T ∗ = ∞ , then we have defined A on R + and the construction is complete. Note that, in either case, a.s. T ∗ = inf { t ≥ | A ( t ) = } . To complete the proof of (i), we need only remark that items 2and 3 of Definition 2.10 are satisfied by the construction of ( σ j ) ∞ j =0 and the abovediscussion.(ii) The idea of the proof is to couple any generative probability measure ¯ µ tothe law of the process A constructed in part (i). Consider a white noise ¯ ξ with anadmissible filtration ( ¯ F t ) t ≥ , a ¯ F -stopping time σ , a solution ¯ A ∈ C ([ s, σ ) , Ω α ) to the SYM driven by ¯ ξ , and a gauge equivalent initial condition A ( s ) = ¯ A ( s ) g ( s ) .Remark that, by part (i) of Theorem 2.8, we can construct on the same probabilityspace a stopping time τ , a time-dependent gauge transformation g ∈ C ([ s, τ ) , G ,α ) (namely g − = ¯ g , the solution to that component of (2.6) driven by ¯ A started withinitial data ¯ g ( s ) = g − ( s ) ) and a solution A ∈ C ([ s, τ ) , Ω α ) to the SYM drivenby the white noise ξ def = Ad g ¯ ξ such that ¯ A g = A on [ s, τ ) . Moreover, by thebound (3.26) in Proposition 3.35, | g | α -Höl cannot blow-up before | ¯ A | α -gr + | A | α -gr does. Since Ω α -gr ֒ → Ω C ,α − (see Section 3.3), and since by Theorem 2.4 we canstart the SYM from any initial condition in Ω C η , η ∈ ( − , ) , it follows that we cantake τ = σ ∧ T ∗ where T ∗ is the blow-up time of | A | α . Note also that g and ξ areadapted to the filtration generated by ¯ ξ , and A is adapted to the filtration generatedby ξ .Consider a, ¯ a ∈ ˆΩ α with [ a ] = [ ¯ a ] and a generative probability measure ¯ µ on D ( R + , ˆΩ α ) with initial condition ¯ a . Let ¯ A ∈ D ( R + , ˆΩ α ) denote the corre-sponding process with filtration ( ¯ F t ) t ≥ , white noise ¯ ξ , and blow-up time ¯ T ∗ asin Definition 2.10. It readily follows from the above remark and the conditions inDefinition 2.10 that there exist, on the same probability space, • a process g : R + → G ,α adapted to ( ¯ F t ) t ≥ , which is càdlàg on the interval [ , ¯ T ∗ ) and remains constant g ≡ on [ ¯ T ∗ , ∞ ) , and • a Markov process A ∈ D ( R + , ˆΩ α ) constructed as in part (i) using the whitenoise ξ def = Ad g ¯ ξ such that A = ¯ A g and A ( ) = a .(Specifically, the process g is constructed to have jumps in [ , ¯ T ∗ ) only at thejump times of ¯ A and A , and ¯ g = g − solves (2.6) driven by ¯ A on its intervals ofcontinuity.) In particular, the pushforwards π ∗ ¯ µ and π ∗ µ coincide, where µ is thelaw of A .To complete the proof, it remains only to show that for the process A frompart (i) with any initial condition a ∈ ˆΩ α , the projected process πA ∈ C ( R + , ˆ O α ) is Markov. However, this follows from the Markov property of A and from taking ymbolic index ¯ µ in the above argument as the law of A with initial condition ¯ a ∼ a . Appendix A Symbolic index
We collect in this appendix commonly used symbols of the article, together withtheir meaning and, if relevant, the page where they first occur.Symbol Meaning Page | · | α Extended norm on Ω k • k ℓ,ε ε -dependent norms on regularity structure of degree ℓ k • k ε , d ε ε -dependent seminorms and metrics on models 101 | • | γ,η,ε ε -dependent norms on modelled distributions 101 A Target space of the jet of the noise and the solution 72 ˜ A − , ˜ A − Negative twisted antipode and its abstract version 71 A A An element of A describing the polynomial part of A ∈ H Cas
Covariance of g valued white noise = quadratic Casimir 88 ¯ C ε , ˆ C ε Renormalisation constants for stochastic YM equation 88 C ε sym , C sym Combination of renormalisation constants and its limit 88 E A generic Banach space 16 F Isomorphism classes of labelled forests 60 F V The monoidal functor between
SSet and
Vec G Compact Lie group 29 G − Renormalisation group 69 g Lie algebra of G G α α -Hölder continuous gauge transformations 29 G ,α Closure of smooth functions in G α H Set of expansions with polynomial part and tree part 76
Hom( s , ¯ s ) Morphisms between two symmetric sets s and ¯ s K ( ε ) Kernel assignment for gauge transformed system 98 ℓ bphz BPHZ renormalisation character 71 M ε The family of K ( ε ) -admissible models 98 Ω Space of additive E -valued functions on X Ω α Banach space { A ∈ Ω | | A | α < ∞} Ω B E -valued -forms with components in a function space B Ω α Closure of smooth E -valued -forms in Ω α O α Space of orbits Ω α / G ,α P ( A ) Powerset of a set A p ∗ Functor from
SSet L to TStruc ¯ L C ∞ ( B ) Space of smooth functions from A to B ymbolic index Symbol Meaning Page ˚ Q (resp. Q ) The set of choices of RHS of SPDE (resp. obeying R ) 72 ̺ Distance function on X R Subcritical, complete rule 64 s A generic symmetric set 47
SSet L The category of symmetric sets with types L TStruc
Category of typed structures, with objects of form Q α ∈A s α h τ i The symmetric set for a labelled rooted tree τ T Isomorphism classes of labelled trees 59 T ( R ) Trees strongly conforming to R T − ( R ) Negative degree unplanted trees in T ( R ) with n ( ̺ ) = 0 T , F Our abstract regularity structures 65 T , F Vector spaces for concrete regularity structure 66 V ⊗ s Symmetric tensor product determined by symmetric set s X Set of line segments 16 Ξ i Symbol for noise, defined as I ( l i , ) ( ) for l i ∈ L − Υ , ¯Υ Maps describing coherence of expansions 79
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