aa r X i v : . [ m a t h . P R ] F e b The Φ measure has sub-Gaussian tails February 24, 2021
Martin Hairer and Rhys Steele
Imperial College LondonEmail: [email protected], [email protected]
Abstract
We provide a very simple argument showing that the Φ measure does have quar-tic exponential tails, as expected from its formal expression. This completes theprogramme of recovering the verification of the Osterwalder and Schrader axiomsfor that measure based purely on SPDE techniques. It has been known since the groundbreaking work of Osterwalder and Schrader[OS73] that, in some “nice” settings, the construction of a (bosonic) quantum fieldtheory satisfying the Wightman axioms is equivalent to the construction of a prob-ability measure on the space of distributions satisfying a number of natural prop-erties. One of the pinnacles of that line of enquiry was the construction in theseventies of the Φ measure [Gli68, EO71, GJ73, FO76, Fel74] (see also [BFS83]for a particularly clean construction), which corresponds to the simplest case of aninteracting theory in three space-time dimensions.In recent years, a new construction of the Φ measure was given by a num-ber of authors [GH18, MW20a], mainly relying on stochastic quantisation [PW81].These “dynamical” constructions have the advantage being able to leverage SPDEtechniques to obtain very fine local properties for the resulting measure. It hasfurthermore been possible to verify all of the Osterwalder–Schrader axioms [GJ12,Sec. 6.1], except for rotation invariance (when considering the whole space) andexponential integrability. The latter property yields analyticity of the correlationfunctions, which is crucial to be able to perform the Wick rotation. Both Gubinelli–Hofmanov´a [GH18] and Moinat–Weber [MW20a] obtain stretched exponential in-tegrability for any exponent strictly less than , which is just short of implying theanalyticity axiom required.At a heuristic level, the Φ measure on the torus T is given up to normalisationby µ ∼ exp (cid:18)Z − |∇ Φ ( x ) | −
14 Φ ( x ) dx (cid:19) Y x ∈ T d Φ ( x ) NTRODUCTION for Φ ∈ S ′ ( T ). The above expression is purely a heuristic, both because the prod-uct measure appearing is not a well-defined object but also because once that mea-sure is successfully interpreted one ends up in a situation where there is need forrenormalisation. Nevertheless, this formal expression strongly suggests that, givenany test function ψ , there exists δ > such that the function Φ exp ( δ h Φ , ψ i ) isintegrable with respect to µ . The goal of this article is to prove precisely this result,which is expected to be optimal based on the formal expression for the measure.Our proof strongly relies on slight modifications of the a priori bounds obtainedin [MW20a], but is otherwise very elementary. As proposed by Parisi and Wu[PW81], we interpret µ as the invariant measure of the Φ equation [Hai14], whichwas shown to exist in [MW17] and is unique by [HM18b, HS19]. Formally, thisequation is given by ∂ t Φ = ∆Φ + ∞ Φ − Φ + ξ , Φ ( , · ) = Φ ( · ) , ( Φ )on the Torus T def = ( R / Z ) for t > where ∆ is the Laplacian on T , ξ is a space-time white noise, and Φ ∈ C α for some α > − is the initial condition. It isknown that for sufficiently small coupling constant, the invariant measure for ( Φ )does indeed coincide with the Φ measure as previously constructed (see [HM18a]).One advantage of the recent constructions however is that they do not rely on anysmallness condition for the coupling constant.Of course, one must correctly interpret the term ∞ Φ appearing in ( Φ ), whichcorresponds to the need for renormalisation for this equation to be well posed. In-deed, even the solution to the linear part of the equation in spatial dimension is aSchwartz distribution rather than a function and hence the cubic term is not a prioriwell-defined. As was shown in [Hai14] (see also [CC18] for an approach via theparacontrolled calculus of [GIP15]), the correct interpretation of a solution to ( Φ )is as the limit in probability as δ → of the solutions to the equations ∂ t Φ ( δ ) = ∆Φ ( δ ) + ( C ( δ ) − C ( δ ) ) Φ ( δ ) − ( Φ ( δ ) ) + ξ δ , Φ ( , · ) = Φ ( · ) , where C ( δ ) i are sequences of diverging constants and ξ δ is the mollification of ξ atscale δ . Whilst the choice of renormalisation constants depends on the choice ofmollifier, the limiting object obtained in this way is independent of the choice ofmollifier. We say that this limit is the solution to ( Φ ).Here we have glossed over the small detail that there is in fact a one-parameterfamily of solutions obtained in this way, since perturbing the renormalisation con-stant by a fixed finite quantity does not affect the convergence result (this parameteris the coupling constant mentioned previously). Since our results apply equally toany element of this one-parameter family, we ignore this detail from here onwards,considering the choice of coupling to be fixed.Since the initial development of a solution theory for ( Φ ), there have been anumber of results establishing various properties for the solution and the associatedsemigroup. Here we mention that in [HM18b], it is established that the semigroup P t associated to ( Φ ) has the strong Feller property. Combining this work with NTRODUCTION [MW20b], one corollary of the results of [HS19] is that this semigroup is alsoexponentially ergodic. A key ingredient for this proof is a powerful a priori boundthat establishes a “coming down from infinity” property for ( Φ ). This kind ofbound was first established via paracontrolled techniques in [MW17] and later amuch shorter argument that is in flavour based on the theory of regularity structureswas given in [MW20a, MW20b]. In particular, it was shown there that if v = Φ − where solves the linear part of the equation then there is some λ > such thatfor fixed t > E [ exp ( λ k v ( t ) k − κ )] < ∞ where the norm is the supremum norm on a subset of the parabolic unit cylindercontaining points at least some fixed distance from the parabolic boundary.In fact, in the verification of this axiom, one is interested in the field testedagainst test functions, rather than the kind of uniform bound given by [MW20b].The main result of this paper is an exponential integrability result which is signifi-cantly stronger than that required by the Osterwalder–Schrader axioms. Theorem 1.1.
Fix κ > sufficiently small and let µ be the invariant measure for ( Φ ) constructed on C − − κ . Let ψ : T → R be a fixed smooth test function anddefine V : C − − κ → R by V ( Φ ) = δ h Φ , ψ i for δ > sufficiently small. Then exp ( V ) ∈ L ( µ ) . The key ingredient of our approach is interpreting the integrability statementas corresponding to finiteness of the measure exp ( V ) dµ . In the same way thatone expects the Φ measure to be invariant for ( Φ ), one expects exp ( V ) dµ to beinvariant for a certain singular SPDE, which we later label ( Ψ ). Hence, as is usualin the program of stochastic quantisation, we proceed to study this measure via theequation ( Ψ ).Without any loss of generality, in what follows we will consider only the casewhere k ψ k ∞ , k Dψ k ∞ ≤ . Remark 1.2.
In principle, one can use the techniques in this paper to also obtainsimilar bounds for other observables so long as no additional renormalisation isrequired and one doesn’t see behaviour of higher than 4th order so that the ad-ditional non-linearity appearing in the corresponding SPDE is controlled by thedampening effect of the cubic term.
Notation and Conventions
Throughout this article we fix the usual parabolic scaling of R = R so thatfor a space-time point z = ( t, x ), k z k = | t | ∨ | x | where | · | is the ℓ ∞ norm.Additionally, we consider R as being equipped with its usual Euclidean scalingand as corresponding to the ‘spatial variables’ in R . NTRODUCTION The scale of regularity of functions in which we will be interested is that of(parabolic) H ¨older spaces. For r > , we let C r = C r ( R d ) be the usual space of r -H ¨older continuous functions. We remark that in the case r ∈ N , this space consistsof ( r − )-times continuously differentiable functions whose ( r − )-th derivativeis Lipschitz continuous rather than the smaller space of r -times continuously dif-ferentiable functions. Further, for r > we denote by B r the set of C r functionswith support in the parabolic ball of radius centered at with C r norm at most .Throughout the article, one should think of r as a sufficiently large fixed integer.For α < , we let C α = C α ( R ) be the space of Schwartz distributions ζ ∈S ′ ( R ) that lie in the dual of the space of compactly supported C r functions for r > −⌊ α ⌋ such that k ζ k α def = sup ϕ ∈B r sup z ∈ R sup λ ∈ ( , ] λ − α |h ζ, ϕ λz i| < ∞ where ϕ λz ( s, y ) = λ − ϕ ( λ − ( s − t ) , λ − ( y − x )) for z = ( t, x ). We adopt a similardefinition for H ¨older spaces of negative regularity over R in which we replace theparabolic scaling with the Euclidean one in the obvious manner.We will fix the values α = − − κ, α ′ = α + κ for κ as in Theorem 1.1. Theimportant feature of this choice is that all results regarding existence of solutionsor convergence of approximations will hold on both C α and C α ′ , allowing us to attimes exploit the compactness of the embedding C α ′ ֒ −→ C α .Finally, for convenience later, we introduce the notation ¯ N def = N ∪ {∞} . In Section 2, we gather the statements of results from later sections that are neces-sary to complete the proof of Theorem 1.1. In this same section we then completesaid proof. The subsequent sections then contain the technical details of adaptingthe required results in the literature to our desired setting. In particular, in Section 3we introduce elements of the theory of regularity structures [Hai14] and their in-homogeneous models, as introduced in [HM18a]. In particular, we show that theequations ( Ψ ,n ) introduced in Section 2 have a solution theory in this frameworkthat yields global in time solutions. In Section 5, we will further show that saidsolutions satisfy a certain a priori bound uniformly in n . Finally, in Section 4 werecall details of the discretisation of regularity structures as found in [HM18a] (seealso [EH19]). The main result of this section is the convergence of a family ofspatially discrete approximations to the solution of ( Ψ ,n ). Acknowledgements
MH gratefully acknowledges support from the Royal Society through a research professor-ship.
ROOF OF T HEOREM As mentioned in the introduction, the main insight of our approach to the proofof Theorem 1.1 is to consider the measure exp ( V ) dµ which one expects to be theinvariant measure for the singular SPDE ∂ t Ψ = ∆Ψ + ∞ Ψ − Ψ + δ h Ψ , ψ i ψ + ξ , Ψ ( , · ) = Ψ ( · ) , ( Ψ )where h· , ψ i is testing in space only.It is not immediate from the constructions of [Hai14] that this equation has asolution theory provided by the framework of regularity structures since the addi-tional nonlinearity appearing is a non-local one. In Section 3, we show that thisadditional nonlinearity poses no serious trouble in building such a solution theoryfor ( Ψ ) using the techniques of regularity structures. Our preferred approach inthis section is that of inhomogeneous models as first presented in [HM18a]. Thisapproach is advantageous both because in the additional nonlinearity time playsa distinguished function-like role and because later we will want to discretise inspace (but not time) again giving the time variable a distinguished role.Combining Theorem 3.7 and Remarks 3.8, 3.9 yields global in time solutionsto equation ( Ψ ). Additionally, these solutions are given as limits in probability as δ → of the pathwise constructed solution to the random PDE( ∂ t − ∆ ) u = − u + ( C δ − C δ ) u + δψ h u, ψ i ψ + ξ δ . (2.1)where C δi are renormalisation constants that diverge as δ → and ξ is the mollifi-cation of space-time white noise at scale δ .This allows us to leverage the techniques used in [MW20b] to prove a prioribounds on the solution to ( Ψ ) that are uniform in the initial condition. If the iden-tification of exp ( V ) dµ as an invariant measure for ( Ψ ) were more than a heuristic,we could conclude the proof of Theorem 1.1 by considering the solution to theequation started from the invariant distribution. Unfortunately, this is not the caseand as a result we will need a priori bounds in a more general setting than statedabove; hence we defer the statement of such bounds until we are in this setting.To overcome this issue, we proceed in two stages of approximation. First wetruncate the additional nonlinearity appearing in equation ( Ψ ). For n ∈ N , let F n : R → R be a smooth function such that F n ( x ) = ( x , | x | ≤ n n + 1 , | x | ≥ n + 1 | F n ( x ) | ∈ [ n , n + 1 ] for | x | ∈ [ n, n + 1 ] and | F ′ n ( x ) | ≤ n for all x ∈ R . Wethen consider the equations (indexed by n ∈ N ) ∂ t Ψ ( n ) = ∆Ψ ( n ) + ∞ Ψ ( n ) − ( Ψ ( n ) ) + δF ′ n ( h Ψ ( n ) , ψ i ) ψ + ξ ( Ψ ,n )with the same initial condition Ψ ( n ) ( , · ) = Ψ ( · ). ROOF OF T HEOREM We additionally let F ∞ ( x ) = x so that equation ( Ψ ) corresponds to ( Ψ ,n )in the case n = ∞ .Equations ( Ψ ,n ) are again formulated in the framework of regularity structuresin Section 3. In Section 5, we adapt the techniques of [MW20b] to prove thefollowing a priori bound which is now uniform in both the initial condition, choiceof ψ and in n . Theorem 2.1.
Fix a function ψ ∈ B r and fix also δ > to be sufficiently small. If Ψ ( n ) is the solution of ( Ψ ,n ) for this ψ , then for all R ∈ ( , ) one has that k Ψ ( n ) k P R ≤ ∨ C max { R − , k k ∞ , − − ; ( , ) , [ τ ] nτ ( − ) | τ | ; τ ∈ L } where L is a collection of trees constructed from the driving noise and P R def = ( R , ) × T . Both this collection of trees and the various norms appearing aredefined in Section 5 In a next stage of approximation, in Section 4, for each n we discretise space toobtain a system of SDEs, labelled ( Ψ ,n ,ε ), approximating ( Ψ ,n ). Simultaneouslywe consider the equivalent discrete approximations of ( Φ ) as considered in e.g.[HM18a, GH18]. The main result of this section is the convergence of the discreteapproximations to the solution of the corresponding continuum equation as the gridscale is sent to .The purpose of these two stages of approximation is as follows. Denoting theinvariant measure of the discrete approximations to ( Ψ ,n ) at grid-scale ε by ν nε and that of the discrete approximation to ( Φ ) by µ ε we have that dν nε = Z − n,ε exp ( F n ( h ι ε · , ψ i )) dµ ε where ι ε : R T ε → C α interprets a function on the discretised Torus T ε def = ε Z ∩ T as a distribution via piecewise constant extension by setting h ι ε F, ϕ i def = X y ∈ T ε Z (cid:3) εy F ( y ) ϕ ( z ) dz where (cid:3) εy def = { z ∈ T : k z − y k ∞ ≤ ε } . In particular, we can exploit the bound-edness of this density to identify that dν n = exp ( F n ( h· , ψ i )) dµ . This knowledge,combined with the a priori bounds of Section 5 allows us to conclude the proof ofTheorem 1.1.The rest of this section will complete the details missing from the remarks inthe above paragraph. Theorem 2.2.
The measures ι ε ∗ µ ε converge weakly on C α to µ as ε → along thedyadics. The same convergence holds for ι ε ∗ ν nε → ν n . ROOF OF T HEOREM Proof.
We begin with the case of ι ε ∗ µ ε . From the results of [GH18], the family ι ε ∗ µ ε is tight (they apply their result to the measures on expanding tori, however theirbounds are all uniform in the length of the torus considered). Hence, it sufficesto show that µ is the unique limit point of the sequence ι ε ∗ µ ε . Then by [EK09,Chapter 4, Theorem 4.5] it suffices to show that if ˜ µ is a weak limit point of ι ε ∗ µ ε then ˜ µ ( f ) = µ ( f ) for all bounded Lipschitz functions f on C α with Lipschitzconstant at most since this set of functions separates points in C α .Fix such a Lipschitz function f on C α . We will show that ι ε ∗ µ ε ( f ) → µ ( f )which is certainly sufficient for our goal.By exploiting invariance, we begin with the simple bound | µ ( f ) − ι ε ∗ µ ε ( f ) | ≤ | µ ( f ) − P t ι ε ∗ µ ε ( f ) | + |P t ι ε ∗ µ ε ( f ) − ι ε ∗ P εt µ ε ( f ) | (2.2)where P t , P εt are the semigroups associated with ( Φ ) and ( Φ ,ε ) respectively.To control the first term in (2.2), we note that the proof of [HS19, Corollary 1.9]shows that P t satisfies the hypotheses of Harris’ Theorem (see e.g. [HM11, Theo-rem 1.2]) and in particular, one even has that k µ − P t ι ε ∗ µ ε k TV → at exponentialrate as t → ∞ .Therefore for fixed η > , we may fix t sufficiently large such that Q def = | µ ( f ) − P t ι ε ∗ µ ε ( f ) | < η uniformly in ε .We now turn to controlling the second term on the right hand side of (2.2) forthis fixed value of t . For this, we write P t ι ε ∗ µ ε ( f ) − ι ε ∗ P εt µ ε ( f ) = Z C α E [ f ( Φ φ ( t )) − f ( ι ε Φ ε p ε φ ( t ))] ι ε ∗ µ ε ( dφ )where Φ φ ( t ) , Φ ε p ε φ ( t )) are the solutions to ( Φ ), ( Φ ,ε ) started from φ and p ε φ )respectively and we have used the fact that p ε def = h· , ε − {k·− x k ∞ ≤ ε } i is a leftinverse to ι ε to identify the appropriate initial condition for the discrete dynamic.By tightness of { ι ε ∗ µ ε : ε = 2 − n , n ≥ } and boundedness of f , there exists acompact set K η such that Q def = Z K cη E [ | f ( Φ φ ( t )) − f ( ι ε Φ ε p ε φ ( t )) | ] ι ε ∗ µ ε ( dφ ) < η . It remains to consider the integral over K η . For this, the crucial remark is that itfollows from Theorem 4.6 that k Φ φ ( t ) − ι ε Φ ε p ε φ ( t ) k C α → as ε → in probabilityuniformly over φ ∈ K η .As a result, there exists an ε ∈ ( , ) such that ε < ε implies that Q def = P (cid:16) ∃ φ ∈ K η such that k Φ φ ( t ) − ι ε Φ ε p ε φ ( t ) k C α ≥ η (cid:17) ≤ η k f k ∞ . ROOF OF T HEOREM Hence, since f is Lipschitz continuous with Lipschitz constant at most , for ε < ε one has the estimate Z K η E [ | f ( Φ φ ( t )) − f ( ι ε Φ ε p ε φ ( t )) | ] ι ε ∗ µ ε ( dφ ) ≤ Q k f k ∞ + η ≤ η . Combining these estimates, we see that |P t ι ε ∗ µ ε ( f ) − ι ε ∗ P εt µ ε ( f ) | ≤ η . Substituting this bound, along with the bound on Q into (2.2) yields that for ε < ε | µ ( f ) − ι ε ∗ µ ε ( f ) | < η . which completes the proof for ι ε ∗ µ ε .It remains to consider ι ε ∗ ν nε . The proof will be the same once we obtain tight-ness and the hypotheses of Harris’ theorem.For tightness, we first note that since ( Ψ ,n ,ε ) is nothing but a system of SDEs, asimple calculation using the generator of this system shows that dν nε = Z − n,ε exp ( F n ( h ι ε · , ψ i )) dµ ε where Z n,ε is a normalising factor. Hence the desired tightness follows immedi-ately from the fact that | F n | is bounded and tightness of { ι ε ∗ µ ε : ε = 2 − n , n ≥ } .It remains to verify the bounds of Harris’ theorem. Here we will only point outthe adaptations needed to [HS19, Corollary 1.9]. The results in Section 5 give therequired ‘coming down from infinity property’ so that all that remains is to see that ν n has full support. This follows from full support of µ [HS19, Theorem 1.8] viaGirsanov transformation. Indeed, from the solution theory of Section 3, if Ψ is thesolution of ( Ψ ,n ) then one can consider the function H n def = F ′ n ( h Ψ , ψ i ) ψ . H n isbounded, smooth in space and η -H ¨older continuous in time for η > sufficientlysmall. The solution to ( Ψ ,n ) then coincides with the solution to the equation ∂ t Ψ ( n ) = ∆Ψ ( n ) + ∞ Ψ ( n ) − ( Ψ ( n ) ) + δH n + ξ. In particular, if ξ is a P -space-time white noise then there exists an equivalentmeasure Q such that H n + ξ is a space-time white noise [All98]. As one wouldexpect, the results of [HM18b, Sections 4 and 5.1] then verify that if Φ is the P -solution for ( Φ ) then under Q , Φ solves the above equation and hence also ( Ψ ,n ).Since P and Q are equivalent measures, this gives the desired result. Remark 2.3.
Whilst in the proof of Theorem 2.2, we essentially control the firstterm on the right hand side of (2.2) in total variation distance and the secondterm in Wasserstein-1 distance, we do not conclude a stronger form of convergencethan weak convergence since on C α , neither of total variation convergence andWasserstein-1 convergence implies the other. ROOF OF T HEOREM
The measure ν n has density Z − n exp ( F n ( h· , ψ i )) with respect to µ .Proof. As noted in the previous proof, dν nε = Z − n,ε exp ( F n ( h ι ε · , ψ i )) dµ ε , hence dι ε ∗ ν nε = Z − n,ε exp ( F n ( h· , ψ i )) dι ε ∗ µ ε . By the weak convergence of ι ε ∗ µ ε , Z n,ε → Z n as ε → . Therefore, bythe same weak convergence, the integrals against Z − n exp ( F n ( h· , ψ i )) dµ and dν n agree on continuous bounded functions and hence the two measures are equal. Lemma 2.5.
The family of measures { ν n : n ∈ N } is tight.Proof. Since the bounds of Section 5 (see Theorem 5.1) are uniform in the initialcondition and in n , for any δ > there is a K such that sup n P ( k Ψ ( n ) ( ) k C α ′ ≥ K ) ≤ δ , where Ψ ( n ) is the solution to ( Ψ ,n ) started from ν n . Since ν n is invariant for thesedynamics, this is nothing but inf n ν n ( B α ′ ( K )) ≥ − δ , where B α ′ ( K ) is the closed ball of radius K in C α ′ . Since the embedding C α ′ → C α is compact, this implies the desired result. Proof of Theorem 1.1.
By Lemma 2.5, there exists a compact set K ⊆ C α suchthat µ ( K ) ≥ and inf n ν n ( K ) = inf n Z − n Z K exp ( F n ( h· , ψ i )) dµ ≥ . Since exp ( F n ( h· , ψ i )) ≤ ∨ exp ( F ∞ ( h· , ψ i )) and exp ( F ∞ ( h· , ψ i )) is continuouson C α , we conclude that sup n exp ( F n ( h· , ψ i )) ≤ C on K for some constant C .Hence ≤ Z − n µ ( K ) C and so Z n ≤ C µ ( K ) . We have that Z n = R C α exp ( F n ( h· , ψ i )) dµ and so the result follows by Fatou’slemma. EGULARITY S TRUCTURES AND I NHOMOGENEOUS M ODELS In this section we recall the definition of a regularity structure and the frameworkof inhomogeneous models as developed in [HM18a]. The significant difference tothe setting of [Hai14] is that in the case of inhomogeneous models, the time vari-able plays a distinguished role and many objects built in the theory are as a resultgenuine functions in time. This set-up is convenient for establishing a solution the-ory for ( Ψ ) since the additional nonlinearity requires the ability to test the solutionat a fixed time against some test function in space. Definition 3.1.
A tuple T = ( T , G ) is a regularity structure if: • T is a graded vector space T = L α ∈A T α , where each T α is a Banachspace and A ⊂ R is a locally finite set. T is called the model space of T . • G is a group of linear transformations of T , such that for every Γ ∈ G , every α ∈ A and every τ ∈ T α one has Γ τ − τ ∈ T <α , with T <α def = L β<α T β . G is called the structure group of T . Remark 3.2.
We have adopted the convention that elements of regularity structuresare coloured blue. This will lend clarity since we will later use a graphical notionin two (similar) ways which will be distinguished by colour. The one exception tothis colouring convention is that functions H : [ , T ) × R d → T won’t be colouredsince it is always clear in which space they are valued. In our setting, we will always work with regularity structures such that each T α is finite-dimensional and A is finite. In particular, there is no ambiguity in thechoice of topology. Assumption 3.3.
Throughout this article, we assume that for a fixed r > allregularity structures T = ( T , G ) contain the structure T poly of polynomials ofscaled degree at most r introduced in [HM18a, Remark 2.2] in the sense that T poly ⊆ T and the restriction of the action of G to T poly coincides with that ofthe group G poly ≃ ( R d +1 , + ) via a group morphism G → G poly . Thus far, the setting described corresponds to the setting of [Hai14] up to thefact that we have insisted on truncating our structures at a fixed maximal homogene-ity. However, in what immediately follows we depart from the original definitionsused there and instead recall the notion of an inhomogeneous model as in [HM18a,Definition 2.4].
Definition 3.4.
For a regularity structure T = ( T , G ) , an inhomogeneous modelis a tuple (( Π tx ) ( t,x ) ∈ R d , ( Γ t ) t ∈ R , ( Σ x ) x ∈ R d ) where • For t ∈ R , Γ t : R d × R d → G , satisfies the algebraic relations Γ txx = 1 , Γ txy Γ tyz = Γ txz , (3.1) for any x, y, z ∈ R d . EGULARITY S TRUCTURES AND I NHOMOGENEOUS M ODELS • For x ∈ R d , Σ x : R × R → G satisfies the algebraic relations Σ ttx = 1 , Σ srx Σ rtx = Σ stx , Σ stx Γ txy = Γ sxy Σ sty , (3.2) for any s, r, t ∈ R and y ∈ R d . • For any ( t, x ) ∈ R d , Π tx : T → S ′ ( R d ) satisfies the algebraic relation Π ty = Π tx Γ txy (3.3) for all y ∈ R d .Additionally, we impose that the actions of Γ txy and Σ stx on T poly are givenby translation by ( , y − x ) and ( t − s, ) respectively, and that the maps Π on T poly ⊆ T are given by (Π tx X ( , ¯ k ) ) ( y ) = ( y − x ) ¯ k , (Π tx X ( k , ¯ k ) ) ( y ) = 0 for k > . Finally, for any γ > and every T > , we assume that there is aconstant C for which the analytic bounds |h Π tx τ , ϕ λx i| ≤ C k τ k λ l , k Γ txy τ k m ≤ C k τ k| x − y | l − m , (3.4a) k Σ stx τ k m ≤ C k τ k| t − s | ( l − m ) / s , (3.4b) hold uniformly over all τ ∈ T l , with l ∈ A and l < γ , all m ∈ A such that m < l ,all λ ∈ ( , ] , all ϕ ∈ B r ( R d ) with r > −⌊ min A⌋ , and all t, s ∈ [ − T, T ] and x, y ∈ R d such that | t − s | ≤ and | x − y | ≤ .If additionally, for δ > the bound |h (Π tx − Π sx ) τ , ϕ λx i| ≤ C k τ k| t − s | δ/ s λ l − δ , (3.5) holds for all τ ∈ T l and the other parameters as before then we say that Π has timeregularity δ > . As is usual, the collection of maps ( Π , Σ , Γ ) as above that satisfy the analyticconstraints but not necessarily the algebraic constraints is a linear space. For anyfixed T > , this space comes equipped with a norm ||| Z ||| γ ; T def = k Π k γ ; T + k Γ k γ ; T + k Σ k γ ; T , where Z = ( Π , Γ , Σ ) and k Π k γ ; T , k Γ k γ ; T and k Σ k γ ; T are the smallest constants C such that the analytic bounds in (3.4a) and (3.4b) hold for the relevant object.In particular, despite not being a linear subspace of this larger space, the spaceof inhomogeneous models inherits a “distance” that is given for a pair of models Z, ¯ Z by ||| Z ; ¯ Z ||| γ ; T def = k Π − ¯Π k γ ; T + k Γ − ¯Γ k γ ; T + k Σ − ¯Σ k γ ; T . (3.6)Additionally, if Π has time regularity δ > then we can account for this by defining k Π k δ,γ ; T def = k Π k γ ; T + C ′ where C ′ is the smallest constant such that the bound (3.5)holds. We then define ||| Z ||| δ,γ ; T and ||| Z, ¯ Z ||| δ,γ ; T analogously to the definitionsabove, replacing all instances of k Π k γ ; T with k Π k δ,γ ; T . EGULARITY S TRUCTURES AND I NHOMOGENEOUS M ODELS
In the following definition, we consider a fixed regularity structure T = ( T , G )with inhomogeneous model Z = ( Π , Γ , Σ ), parameters γ, η ∈ R , a time T > ,and H : ( , T ] × R d → T <γ . We define the quantity k H k γ,η ; T def = sup t ∈ ( ,T ] sup x ∈ R d sup l<γ | t | ( l − η ) ∨ k H t ( x ) k l + sup t ∈ ( ,T ] sup x = y ∈ R d | x − y |≤ sup l<γ k H t ( x ) − Γ txy H t ( y ) k l | t | η − γ | x − y | γ − l , (3.7)where l ∈ A in the third supremum and | t | def = | t | ∧ . This quantity is a partialanalogue of the homogeneous D γ,η norm introduced in [Hai14]. However it doesnot account for any of the behaviour of H in time and as a result, the restrictionin the second supremum that | x − y | ≤ | t, s | def = | t | ∧ | s | appearing in thehomogeneous case has been removed. This turns out to be important for obtainingthe relevant Schauder estimates; see [HM18a, Theorem 2.21].We then define the inhomogeneous D γ,ηT norm as ||| H ||| γ,η ; T def = k H k γ,η ; T + sup s = t ∈ ( ,T ] | t − s |≤| t,s | s sup x ∈ R d sup l<γ k H t ( x ) − Σ tsx H s ( x ) k l | t, s | η − γ | t − s | ( γ − l ) / s . (3.8) Definition 3.5.
We define D γ,ηT ( Z ) to be the space of functions H : ( , T ] × R d →T <γ such that ||| H ||| γ,η ; T < ∞ . Remark 3.6.
Inhomogeneous analogues of the usual Reconstruction Theorem andSchauder estimates hold true in this setting; see [HM18a, Theorems 2.11 and 2.21].Formulating a suitable fixed point result requires a little more care than in thehomogeneous case, but this is also obtained in [HM18a]. The subject of the nextsubsection is a formulation of their construction specific to our setting.
Given a second model ¯ Z = ( ¯Π , ¯Γ , ¯Σ ) for T , we define the distance ||| H ; ¯ H ||| γ,η ; T between H ∈ D γ,ηT ( Z ) and ¯ H ∈ D γ,ηT ( ¯ Z ) by setting k H ; ¯ H k γ,η ; T def = sup t ∈ ( ,T ] sup x ∈ R d sup l<γ | t | ( l − η ) ∨ k H t ( x ) − ¯ H t ( x ) k l + sup t ∈ ( ,T ] sup x = y ∈ R d | x − y |≤ sup l<γ k H t ( x ) − Γ txy H t ( y ) − ¯ H t ( x ) + ¯Γ txy ¯ H t ( y ) k l | t | η − γ | x − y | γ − l , ||| H ; ¯ H ||| γ,η ; T def = k H ; ¯ H k γ,η ; T + sup s = t ∈ ( ,T ] | t − s |≤| t,s | s sup x ∈ R d sup l<γ k H t ( x ) − Σ tsx H s ( x ) − ¯ H t ( x ) + ¯Σ tsx ¯ H s ( x ) k l | t, s | η − γ | t − s | ( γ − l ) / s . EGULARITY S TRUCTURES AND I NHOMOGENEOUS M ODELS ( Φ )It was shown in [Hai14, Section 8.1] that given a locally subcritical equationwith nonlinearity of the form F ( u, ∇ u, ξ ), one can construct a regularity structure T = ( T , G ) and an associated space of modelled distributions such that the givenequation can be formulated as a fixed point problem is that space.The equation ( Ψ ,n ) does not quite fit immediately into that framework sinceits nonlinearity contains the non-local term δ h Ψ ( n ) , ψ i ψ . Instead, we consider theregularity structure T constructed in [Hai14] for ( Φ ) and suitably interpret thenon-local term of ( Ψ ,n ) in this regularity structure, allowing us to work with thatfixed regularity structure for both equations.We don’t recall here the full details of the construction of [Hai14]. Instead, wemention only that the construction includes the recursive construction of sets ofsymbols F , U that one would expect to need to formulate the fixed point argumentfor ( Φ ).The set F contains the symbols required to describe terms appearing on theright hand side of equation ( Φ ). In particular, F is a subset of the set of symbolsgenerated from { , X i , Ξ } under the operations τ
7→ I ( τ ) and ( τ , ¯ τ ) τ ¯ τ andthe assumptions that the multiplication is commutative with identity element andthat I ( X k ) = 0 for each k ∈ N . In the case of ( Φ ), the first few symbols of F are F = { , Ξ , , , , X i , , , , , . . . } . Here we have adopted the usual graphical notation of defining def = I ( Ξ ) andthen defining the remaining rooted trees recursively, where abstract integration isrepresented by drawing an edge downward from the root and multiplication is rep-resented by concatenation of trees at the root.Meanwhile, U contains the symbols required to describe the solution of equa-tion ( Φ ). Concretely, we have that U = { X k : k ∈ N } ∪ I ( F ). In the caseof ( Φ ), we have that U ⊆ F . The regularity structure T then has model space T = span F equipped with the grading | · | defined by | | = 0 , | X i | = 1 , | Ξ | = − − κ, |I ( τ ) | = | τ | +2 , | τ ¯ τ | = | τ | + | ¯ τ | where κ > is chosen sufficiently small. At this point, the construction givenin [Hai14, Section 8.1] yields a structure group G acting on T with the desiredproperties. We will not give details of the construction here but rather will recallthe key properties as we need them.Since we aim to formulate our equations as fixed point problems in a D γ,ηT space, we will not need the symbols beyond a suitably large homogeneity. Hence,for the rest of the paper we will work in the ambient regularity structure T r withmodel space T To complete this section we formulate ( Ψ ,n ) in the setting described earlier in thissection. We remark that the equivalent program for ( Φ ) was already completed in[HM18a]. The modifications for ( Ψ ,n ) are only minor, but since the nonlinearityis non-local we will briefly outline what needs to be done to accommodate it.Defining the maps ˆ F ( τ ) = − Q ≤ ( τ ) , I = − , the abstract analogue of ( Φ ) is given as in [HM18a] by the fixed point problem U = P ˆ F ( U ) + S Φ + I , (3.10)where P def = K ¯ γ + R γ R is the usual lift of the action of the heat kernel and S Φ denotes the lift of the solution of the heat equation with initial condition Φ tothe polynomial part of the regularity structure. We refer the reader to equations(2.29), (3.2) and Theorem 2.11 of [HM18a] for more details on the definitions ofthe operators appearing on the right-hand side of this equation. Formulating the EGULARITY S TRUCTURES AND I NHOMOGENEOUS M ODELS fixed point problem in this way is advantageous since ˆ F has the property that if T U = span U then ˆ F ( T U ∩ ˆ T ) ⊆ ˆ T .In order to formulate the corresponding analogue of ( Ψ ,n ), we introduce for n ∈ ¯ N the maps G n : D γ,ηT → D γ,ηT given by G n ( U ) = ( hR t U, ψ i ψ n = ∞ F ′ n ( hR t U, ψ i ) ψ n < ∞ where R t is the reconstruction operator at time t – see [HM18a, Thm 2.11]. Herewe begin to reap the benefits of working in the inhomogeneous setting since hR t U, ψ i is automatically well defined even though the testing is only in space; this wouldnot be automatic in the setting of [Hai14].We then define the abstract analogue of ( Ψ ,n ) ( n ∈ ¯ N ) to be U n = P ( ˆ F ( U n ) + G n ( U n )) + S Φ + I . (3.11)In this setting, one has the following analogue of [HM18a, Thm 3.10]. Theorem 3.7. Let γ = 1+ κ and η ≤ α . Then for any model Z with time regularity δ > on ˆ T and for every periodic Φ ∈ C η ( R d ) , there exists a time T ∗ ∈ ( , ∞ ] such that for every T < T ∗ the equations (3.10) , (3.11) admit a unique periodicsolution U ∈ D γ,ηT ( Z ) for all n ∈ ¯ N . Furthermore, if T ∗ < ∞ then either lim T ↑ T ∗ kR T S Φ T ( Φ , Z ) T k C η = ∞ or there is an n ∈ ¯ N such that lim T ↑ T ∗ kR T S Ψ n T ( Φ , Z ) T k C η = ∞ where S Φ T , S Ψ n T : ( Φ , Z ) U are the solution maps for (3.10) and (3.11) respec-tively.Additionally, for every T < T ∗ , the solution maps S Φ T , S Ψ n T are jointly Lipschitzcontinuous in a neighbourhood around ( Φ , Z ) in the sense that, for any B > there is C > such that, if ¯ U = S Φ T ( ¯Φ , ¯ Z ) for some initial data ( ¯Φ , ¯ Z ) where ¯ Z has time regularity δ > , then one has the bound ||| U ; ¯ U ||| γ,η ; T ≤ Cδ , provided k Φ − ¯Φ k C η + ||| Z ; ¯ Z ||| γ ; T ≤ δ , for any δ ∈ ( , B ] and similarly for S Ψ n T .Proof. The result for the case of ( Φ ) follows almost exactly as in the fixed pointargument of [Hai14] (see Theorem 7.8 and Proposition 7.11 there) and is the con-tent of [HM18a, Thm 3.10]. Since we have a direct interpretation of the additionalnonlinearity of ( Ψ ,n ) in the polynomial part of our regularity structure, the resultfor that equation will follow from the same techniques with no difficulties as soonas we verify that G n satisfies the bound ||| G n ( H ) , G n ( ¯ H ) ||| ¯ γ,η − ¯ γ ; T . ||| H, ¯ H ||| γ,η ; T + ||| Z, ¯ Z ||| γ ; T for all n ∈ ¯ N for some < ¯ γ ≪ δ . Since ψ ∈ B r , this is immediate from theregularity of the reconstruction operator provided by the bound (2.13) of [HM18a,Thm 2.11], combined with smoothness of F ′ n and x x . ISCRETE I NHOMOGENEOUS M ODELS If the model Z in the preceding theorem is the canonical lift of asmooth driving noise as in [Hai14, Sec. 8.2] then the reconstruction operator isgiven by ( R t U t ) ( x ) = (cid:0) Π tx U t ( x ) (cid:1) ( x ) . In this case, the reconstruction of the abstract solutions found above coincides withthe classical solutions of ( Φ ) , ( Ψ ) with smooth driving noise and no renormali-sation.Additionally, if we define the renormalisation map M as in [Hai14, Sec. 9.2]with constants C and C and build a renormalised smooth model Z M as in[Hai14, Sec. 8.3], then u Φ def = R t S Φ t ( Φ , Z ) t and u Ψ n def = R t S Ψ n t ( Ψ , Z ) t solvethe following equations with smooth driving noise (cf. [Hai14, Prop. 9.10]) ∂ t u Φ =∆ u Φ − u + ( C − C ) u Φ + ξ,∂ t u Ψ n =∆ u Ψ n − u n + ( C − C ) u Ψ n + δF ′ n ( h u Ψ n , ψ i ) ψ + ξ. Finally, if ξ ε = ̺ ε ∗ ξ where ̺ ε is a mollifier at scale ε , then there is a choice ofdiverging constants C ε , C ε such that the solutions to the above equations convergein probability as ε → (cf. [CH16]). We define the solution of ( Φ ) , ( Ψ ,n ) to bethese limits which are independent of the choice of mollifier. Remark 3.9. It is known that ( Φ ) has a ‘coming down from infinity’ propertythat precludes the blow-up of C α norms of solutions (cf. [MW20b]). Section 5adapts the techniques of [MW20b] to show the corresponding result for ( Ψ ,n ) . Asa result, in the cases of interest for us it follows that in the setting of Theorem 3.7one actually has T ∗ = ∞ . In this section we introduce the discrete analogues of the objects and the resultsof the last section. We will use the discretisations to identify the density of theinvariant measure of ( Ψ ,n ) with respect to ( Φ ) for n ∈ N . In particular, the goalis to treat for an arbitrary fixed n ∈ N discretisations of ( Φ ), ( Ψ ,n ) of the form ddt Φ ε =∆ ε Φ ε + C ( ε ) Φ ε − (Φ ε ) + ξ ε , Φ ε ( , · ) = Φ ε ( · ) , ( Φ ,ε ) ddt Ψ ε =∆ ε Ψ ε + C ( ε ) Ψ ε − (Ψ ε ) + δF ′ n ( h ι ε Ψ ε , ψ i ) ψ ε + ξ ε , Ψ ε ( , · ) = Ψ ε ( · ) , ( Ψ ,n ,ε )on the discretisation T ε of T with grid scale ε = 2 − N for N ∈ N , where Φ ε , Ψ ε ∈ R T ε , ∆ ε is the nearest-neighbour approximation of the Laplacian ∆ and ξ ε arespatial discretisations of ξ defined on a single common probability space by setting ξ ε ( t, x ) def = ε − h ξ ( t, · ) , (cid:3) εx i , ( t, x ) ∈ R × T ε , (4.1)where, for x ∈ T ε , (cid:3) εx ⊂ T denotes the cube of side length ε centred at x . ISCRETE I NHOMOGENEOUS M ODELS The function ψ ε ∈ R T ε is defined by ψ ε ( y ) = ε − Z (cid:3) εy ψ ( z ) dz. Finally C ( ε ) λ ∼ ε − + log ε is a sequence renormalisation constants for which pre-cise values are given in [HM18a, Eq. 7.6] and the subsequent paragraph.Of course, equations ( Φ ,ε ) and ( Ψ ,n ,ε ) are nothing but SDEs with global in timesolutions. However, in order to prove the convergence to the continuum solutions,it is useful to recast their solution theory in the language of regularity structures.We recall the following definitions from [HM18a]. Definition 4.1. Given ε > , a discrete model at grid-scale ε for the regularitystructure T consists of maps ( Π ε , Γ ε , Σ ε ) Π ε,tx : T → R T ε , Γ ε,t : T ε × T ε → G , Σ εx : R × R → G , indexed by t ∈ R and x ∈ T ε , which have all the algebraic properties of theircontinuous counterparts in Definition 3.4, with the spatial variables restricted tothe grid T ε . Additionally, we assume that (Π ε,tx τ ) ( x ) = 0 , for all τ ∈ T l with l > ,and all x ∈ T ε and t ∈ R . The seminorms k Π ε k ( ε ) γ ; T and k Γ ε k ( ε ) γ ; T are defined to be the smallest constantssuch that the inequalities (3.4a) hold uniformly in λ ∈ [ ε, ], x, y ∈ T ε , t ∈ R and with the usual duality pairing of D ′ ( R × T ) × D ( R × T ) replaced with thediscrete pairing h F, ϕ i ε def = Z R X y ∈ T ε F ( t, y ) ϕ ( t, y ) dt. The quantity k Σ ε k ( ε ) γ ; T is then defined as the smallest constant C such that thebounds k Σ ε,stx τ k m ≤ C k τ k ( | t − s | / s ∨ ε ) l − m , (4.2)hold uniformly in x ∈ T ε and the other parameters as in (3.4b).We measure the time regularity of Π ε as in (3.5), by substituting the continuousobjects by their discrete analogues, and by using | t − s | / s ∨ ε instead of | t − s | / s on the right-hand side. We also define quantities k · k ( ε ) , ||| · ||| ( ε ) , in the same way asthe above construction that measure the size of (resp. distance between) model(s) Z (resp. Z, ¯ Z ). Remark 4.2. The pairing h· , ·i ε does not correspond to the embedding ι ε . Indeed,it does not correspond to any embedding e : R T ε → C − − since the action inspace is that of a Dirac delta which has regularity no better than − . This is nota serious issue for us since the difference between the two ways of testing appliedwith the solutions of ( Φ ,ε ) , ( Ψ ,n ,ε ) converges to as ε → . ISCRETE I NHOMOGENEOUS M ODELS One then has the following discrete analogue of the D γ,ηT spaces. For γ, η ∈ R , a fixed time T > and a discrete model Z ε = ( Π ε , Γ ε , Σ ε ) on a regularitystructure T , we define the D γ,ηT,ε norms k H ε k ( ε ) γ,η ; T and ||| H ε ||| ( ε ) γ,η ; T of a function H ε : ( , T ] × T ε → T <γ in exactly the same way as in (3.7) and (3.8), except thatthe spatial variables run over T ε and the powers of | t | and | t, s | appearing thereare replaced by | t | ∨ ε and | t, s | ∨ ε respectively. Definition 4.3. D γ,ηT,ε is the space of functions H ε : ( , T ] × T ε → T <γ such that ||| H ε ||| ( ε ) γ,η ; T < ∞ . Remark 4.4. In the setting of discrete inhomogeneous models, suitable instancesof the usual results in the theory of regularity structures hold. For example, one hasa reconstruction operator with the explicit representation ( R εt H εt )( x ) def = ( Π ε,tx H εt ( x ))( x ) [HM18a, Thm 4.6]. Additionally, the Green’s function for the discretised heatequation has a decomposition that is a suitable analogue of the decomposition ofthe heat kernel given in [Hai14] (see [HM18a, Lem. 5.4]) and the correspondinglift of the action of the kernel satisfies analogues of the usual Schauder estimates,[HM18a, Thm 4.17]. We now obtain the solutions of ( Φ ,ε ), ( Ψ ,n ,ε ) from an abstract fixed point ar-gument. We will eventually handle the renormalisation terms containing factors C ( ε ) λ at the level of our choice of model so that the abstract formulation of theseequations will be given by U ε = P ε ˆ F ( U ε ) + S ε Φ ε + I (4.3) U εn = P ε ( ˆ F ( U εn ) + G εn ( U εn )) + S ε Φ ε + I (4.4)where S ε Φ ε is the solution to the semidiscrete heat equation with initial condi-tion Φ ε ∈ R T ε and P ε = K ε ¯ γ + R εγ R ε is the abstract analogue of the action ofthe semidiscrete heat kernel (see equations ( . ) and ( . ) of [HM18a] for thedefinitions of R εγ and K ε ¯ γ ). Finally G εn ( U ε )( t, x ) def = ( F ′ n ( h ι ε R εt U ε , ψ i ) ψ ε ( x ) , n ∈ N h ι ε R εt U ε , ψ i ψ ε ( x ) , n = ∞ One then has the following analogue of Theorem 3.7, which is essentially specialcase of [HM18a, Thm 5.8] up to the minor adaptation required to accommodatethe nonlinearity G εn which is similar to that performed in Section 3 and so we omitthe details. Theorem 4.5. Let Z ε be a sequence of discrete inhomogeneous models indexed by ε = 2 − N for N ≥ . Then for every T < ∞ , the sequence of solution maps S εT : ( Φ ε , Z ε ) U ε of the equation (4.3) up to time T is jointly Lipschitz continuous(uniformly in ε ) in the sense of Theorem 3.7, but replacing the continuum objectswith their discrete analogues. The same is true of the solution map for (4.4) . ISCRETE I NHOMOGENEOUS M ODELS In order to state our direct analogue of the main convergence result of [HM18a]we introduce a choice of discretisation of initial condition φ ∈ C α ( T ). We set p ε φ def = h φ, ε − {k·− x k ∞ ≤ ε } i ∈ L ∞ ( T ε ) . The right hand side of this expression is well defined for φ ∈ C − κ since theindicator function of a cube lies in the Besov space B − κ , . A simple calculationusing the Littlewood-Paley decomposition then shows that if ε = 2 − k then theoperator norm of p ε : C α → L ∞ ( T ε ) is bounded uniformly in k ≥ . Theorem 4.6. Let ξ be a space-time white noise over L ( R × T ) on a probabilityspace ( Ω , F , P ) and let Φ and Ψ be the unique maximal solution of ( Φ ) and ( Ψ ,n ) respectively with initial condition φ ∈ C α ( T ) . Let ξ ε be given by (4.1) ,and let Φ ε and Ψ ε be the unique global solution of ( Φ ,ε ) and ( Ψ ,n ,ε ) respectivelywith initial condition p ε φ . Then there exists a sequence of stopping times T ε suchthat P ( T ε < T ) → as ε → for any fixed T positive and sup t ∈ [ ,T ε ] k Φ ( t ) − ι ε Φ ε ( t ) k C − − → , sup t ∈ [ ,T ε ] k Ψ ( t ) − ι ε Ψ ε ( t ) k C − − → , in probability as ε → . Furthermore the above convergence is locally uniform inthe initial condition φ .Proof. This follows from the same techniques as the proof of [HM18a, Thm 1.1]for the case of ( Φ ). This proof proceeds by using Theorem 3.7 and Theorem 4.5along with convergence of suitable Gaussian models to reduce to the convergencein the case of a smooth driving noise which is a problem of numerical analysis.We note that the same techniques work here since we have formulated the ab-stract version of ( Ψ ,n ) and its discrete counterpart on the same regularity structureas those for ( Φ ) such that the equations are simultaneously driven by the samechoice of model. Hence following the proof of [HM18a, Thm 1.1] yields the con-vergence sup t ∈ [ ,T ε ] (cid:16) k Φ ( t ) ; Φ ε ( t ) k ( ε ) C − − ∨ k Ψ ( t ) ; Ψ ε ( t ) k ( ε ) C − − (cid:17) → , in probability as ε → , where k ζ ; ζ ε k ( ε ) C α def = sup ϕ ∈B r sup x ∈ T ε sup λ ∈ [ ε, ] λ α |h ζ, ϕ λx i−h ζ ε , ϕ λx i ε | and T ε is a suitable subsequence of H K ∨ ˜ H K where H K (resp. ˜ H K ) isthe exit time of the ball of radius K for Φ (resp. Ψ ).Unfortunately, as mentioned earlier, the discrete testing appearing here is thewrong one. Since the behaviour at below scale ε is straightforward, it remains tosee that sup ϕ ∈B r sup x ∈ T ε sup λ ∈ [ ε, ] λ α |h ι ε Φ ε , ϕ λx i − h Φ ε , ϕ λx i ε | → and similarly for Ψ ε . Notice that we can write h ι ε Φ ε , ϕ λx i − h Φ ε , ϕ λx i ε = X y ∈ T ε Φ ε ( y ) Z (cid:3) εy (cid:16) ϕ λx ( z ) − ϕ λx ( y ) (cid:17) dz . λ − ε k Φ ε k ∞ OUNDS FOR THE C ONTINUUM Ψ ,n EQUATION since | ϕ λx ( y ) − ϕ λx ( z ) | ≤ λ − | z − y | and the summand is non-zero only for y suchthat | y − x | ≤ λ . Hence, it suffices to see that k Φ ε k ∞ . ε α .For this we only sketch the details, since they are an application of the samediscrete tools as used repeatedly in this paper and in [HM18a]. Indeed, this isa corollary of the same rate of blow up for discretisations of the stochastic heatequation since as usual for ( Φ ), Φ = u + v where u is the solution of the stochasticheat equation and v ∈ C − κ is the solution of a ‘remainder equation’. One has asimilar decomposition for Φ ε and the techniques used above yield that sup t ∈ [ ,T ε ] k v ε ( t ) − v ( t ) k ∞ → for some sequence of stopping times T ε satisfying P ( T ε < T ) → as ε → forevery fixed T > . In particular, taking T ε = T ε ∧ T ε yields the desired resultsince then the above yields control on the supremum norm of v ε so that the onlyblow-up in Φ ε comes from u . Ψ ,n equation In this short section, we state an a priori bound which is uniform in n and ε for thePDE with smooth driving noise( ∂ t − ∆ ) u = − u + ( C ( ε ) − C ( ε ) ) u + δψF ′ n ( h u, ψ i ) + ξ ε (5.1)that is a direct adaptation of the main result of [MW20b] which give the equivalentbound for the solution of ( Φ ). (Take for example ξ ε = ̺ ε ⋆ ξ for ̺ ε a smoothcompactly supported mollifier.) The constants C ( ε ) i are the same renormalisationconstants as in the proof of [Hai14, Thm 10.22]; in particular they do not dependon the additional nonlinearity appearing on the right-hand side. This kind of boundis of interest to us since the terms appearing on the right-hand side of our a prioribound will converge to natural limiting objects as ε → so that these bounds willdirectly transfer to the solution of ( Ψ ,n ).Since we are able to restart the equation, it is sufficient for us to obtain goodbounds up to time . Hence we define the cylinder P def = ( , ) × T and for R > we set P R = ( R , ) × T . The bounds we obtain will also depend on R in an explicit way which will enable the bounds to be independent of the choice ofinitial condition.If is a solution of the equation ( ∂ t − ∆ ) = ξ ε then the techniques of [MW20b]will in fact yield supremum norm bounds on v def = u − where u is the solution of(5.1). However, since we will only use these bounds to gain control on the C α norm of u ( t, · ), we simply state the bound in this form.First, we introduce the definitions of some graphical notation appearing in[MW20b]. We emphasise that the trees appearing in this section are coloured blacksince they are not elements of a regularity structure. They also slightly differ from OUNDS FOR THE C ONTINUUM Ψ ,n EQUATION the BPHZ model applied to those elements since they are constructed directly fromthe PDE, rather than via convolution with some cut-off version of the heat kernel.We define def = − C ( ε ) and def = − C ( ε ) , leaving the ε -dependenceimplicit. We then introduce the higher order symbols , which are assumed tosatisfy ( ∂ t − ∆ ) = , ( ∂ t − ∆ ) = . For α > , we let [ · ] α,C be the usual α -H ¨older seminorm restricted to pointsin the set C ⊆ R × T . If C is omitted, it is to be understood that it is the wholespace.To define an analogue of these seminorms for α < , we fix a family of smoothcompactly supported test functions φ T : R × R → R with a semigroup propertyat dyadic scales as constructed in [MW20b, Sec. 2]. The precise form of φ T won’tmatter to us except that it is required to prove the analogue of the reconstructiontheorem [MW20b, Thm 2.8] used in the paper of Moinat and Weber which is im-plicitly also required here. Since we do not retrace many details of the proofsof [MW20b] in this section, we refer the interested reader to that paper for moredetails.Having introduced this quantity, we now introduce a finite collection T = { , , x , , , , , , } , of higher order trees. Each of these trees represents an ε -dependent random func-tion (it actually depends furthermore on a space-time “base point” since we con-sider the “positively renormalised” quantities). Details of their construction do notmatter for the purpose of this discussion, but we introduce the quantity n τ countingthe number of leaves of a tree τ (so n = 2 , n = 4 , etc) as well as a (random)quantity [ τ ] κ measuring the size of these functions in a terms of how their convo-lution with φ T behaves at the base point as T → . (If the functions are base pointindependent, as is the case for example for and , then these are equivalent to aH ¨older norm of order deg τ − κ , where deg = − , the degree is multiplicative,and solving the heat equation increases degree by .) See [MW20b, Eq. 2.13–2.19]for details of these definitions (note that their integer multiples of ε are replaced by κ in our notation). One has for example[ ] κ = sup x ∈ P sup T < T κ (cid:12)(cid:12)(cid:12)(cid:12)Z ( y ) Ψ T ( y − x ) dy (cid:12)(cid:12)(cid:12)(cid:12) , [ ] κ = sup x ∈ P sup T < T κ (cid:12)(cid:12)(cid:12)(cid:12)Z (cid:16) ( y ) ( y ) − C ( ε ) − ( x ) ( y ) (cid:17) Ψ T ( y − x ) dy (cid:12)(cid:12)(cid:12)(cid:12) . Finally, since the tree naturally plays a distinguished role in the equation for u − because of the non-local term in the nonlinearity, we measure its regularityin a slightly stronger norm than [MW20b] in order to get good bounds, and we set[ ] κ = sup t ∈ [ , ] k ( t, · ) k C − − κ ( T ) . OUNDS FOR THE C ONTINUUM Ψ ,n EQUATION Fix a smooth function ψ : T → R such that k ψ k ∞ , k Dψ k ∞ ≤ and fix also δ > and κ > small enough. If u is the solution of (5.1) for this ψ and v = u − , then for all R ∈ ( , ) one has the bound k v k P R ≤ C max n R − , [ τ ] nτ ( − κ ) κ ; τ ∈ T o , (5.2) where n τ is the number of leaves appearing in τ . Here C is a constant that isindependent of n and ε , and k v k P R denotes the supremum norm over P R .Proof. This follows with only very minor modifications of the proof of [MW20b,Thm 2.1]. Indeed, one only has to make adjustments to deal with the extra termappearing in the nonlinearity of our equation. This only requires small changes inSection 4.2 of that paper since, once one derives a similar bound to that given inthe conclusion of that subsection, one can proceed with the rest of the proof withno significant changes.The structure of the proof there is to assume that the bound k v k P R ≤ ∨ C max n [ τ ] nτ ( − κ ) κ ; τ ∈ T o , (5.3)fails on some parabolic cylinder with a constant C that depends only on combinato-rial factors arising during their proof and then derive from the converse inequalitya bound of order R − , thus yielding (5.2).There are only two steps in the proof which rely on the precise form of theequation under consideration and not just on the local expansion of the solution upto order (which has the precise same form for ( Ψ ,n ) as for ( Φ )). The first step,which is given in their Section 4.2, is to consider the two-parameter function U given by U ( x, y ) = v ( y ) − v ( x ) + ( y ) − ( x ) + 3 v ( x ) ( ( y ) − ( x ) ) , see [MW20b, Eq. 2.29], to fix an open space-time domain D = P R (for some R > ), and to assume a bound of the type [ τ ] κ ≤ c k v k n τ ( − κ ) / D for some c ≤ .Writing ( · ) T for space-time convolution with φ T , one then shows that there exists β ≥ such that a bound of the form T k ( ( ∂ t − ∆ ) U ( x, · ) ) T k B ( x,L ) . k v k D ( L/T ) β , (5.4)holds uniformly over all choices of D with diameter bounded by , all x ∈ D , andall T ≤ L ≤ ∧ k v k − D such that B ( x, L ) ⊂ D , where B ( x, L ) denotes the parabolic ball of radius L “directed towards the past”, see [MW20b, Eq. 2.2]. Remark 5.2. The bound given in [MW20b, Eq. 4.20] appears stronger because ofthe presence of the constant c which can be made arbitrarily small. That boundhowever is incorrect since the first term in [MW20b, Eq. 4.8] does not satisfy it.Fortunately, this additional factor c is not exploited in the sequel. OUNDS FOR THE C ONTINUUM Ψ ,n EQUATION The only difference between ( ∂ t − ∆ ) U in the case of ( Ψ ,n ) compared to thatof ( Φ ) is that we obtain an additional term δF ′ n ( h v + , ψ i ) ψ , which is easilybounded by δ k F ′ n ( h v + , ψ i ) ψ k D ≤ δ kh v + , ψ i ψ k D . X j =0 kh v, ψ i j k P R kh , ψ i − j k D . k v k D + [ ] κ . T − k v k D , with constants uniform in n . Here, we made use of the bounds T ≤ ∧ k v k − D and [ ] κ . k v k / − κD ≤ k v k D , where the last inequality follows from the fact that(5.3) is assumed to fail. 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