aa r X i v : . [ m a t h . P R ] M a r Rough flows
I. BAILLEUL and S. RIEDEL Abstract.
We introduce in this work a concept of rough driver that somehow provides arough path-like analogue of an enriched object associated with time-dependent vector fields.We use the machinery of approximate flows to build the integration theory of rough driversand prove well-posedness results for rough differential equations on flows and continuity ofthe solution flow as a function of the generating rough driver. We show that the theoryof semimartingale stochastic flows developed in the 80’s and early 90’s fits nicely in thisframework, and obtain as a consequence some new strong approximation results for generalsemimartingale flows and provide a fresh look at large deviation theorems for ’Gaussian’stochastic flows.
Introduction
An elementary construction recipe of flows was recently introduced in [Bai15] and usedthere to get back the core results of Lyons’ theory of rough differential equations [Lyo98,FV10] in a very short and elementary way. This work emphasizes the fact that it maybe worth considering flows of maps as the primary objects from which the individualtrajectories can be built, as opposed to the classical point of view that constructs a flowfrom an uncountable collection of individual trajectories. Probabilists know how trickyit can be to deal with uncountably many null sets. As well-recognized now, the mainsuccess of Lyons’ theory was to disentangle probability from pure dynamics in the studyof stochastic differential equations by showing that the dynamics is a deterministic andcontinuous function of an enriched signal that is constructed from the noise in the equationby purely probabilistic means. This very clean picture led to new proofs and extensions offoundational results in the theory of stochastic differential equations, such as Stroock andVaradhan support theorem or the basics of Freidlin and Ventzel theory of large deviationsfor diffusions.It was realized in the late 70’s that stochastic differential equations not only defineindividual trajectories, they also define flows of regular homeomorphisms, depending onthe regularity of the vector fields involved in the dynamics [Elw78, Bis81, IW81, Kun81b].This opened the door to the study of stochastic flows of maps for themselves [Har81,Bax80, LJ82, LJ85] and it did not take long time before Le Jan and Watanabe [LJW84]clarified definitely the situation by showing that, in a semimartingale setting, there is aone-to-one correspondence between flows of diffeomorphisms and time-varying stochasticvelocity fields, under proper regularity conditions on the objects involved. We offer in thepresent work an embedding of the theory of semimartingale stochastic flows into the theoryof rough flows similar to the embedding of the theory of stochastic differential equationsinto the theory of rough differential equations. While the acquainted reader will havenoticed that the latter framework can be used to deal with Brownian flows by seeing them I.B. was partly supported by the ANR project ”Retour Post-doctorant”, no. 11-PDOC-0025; I.B. alsothanks the U.B.O. for their hospitality, part of this work was written there. as solution flows to some Banach space-valued rough differential equations driven by aBrownian rough path in some space of vector fields, such as done in [Der10, DD12], thesituation is not so clear for more general random velocity fields and stochastic flows ofmaps. Our approach provides a simple setting for dealing with the general case, and weprovide in this work an elementary direct approach to the construction of stochastic flowswhose scope goes beyond the realm of semimartingale calculus.Our theory of rough flows is based on the deterministic "Approximate flow-to-flow"machinery introduced in [Bai15], which gives body to the following fact. To a 2-indexfamily ( µ ts ) s t T of maps which falls short from being a flow one can associate a uniqueflow ( ϕ ts ) s t T close to ( µ ts ) s t T ; moreover the flow ϕ depends continuously on theapproximate flow µ . We introduce a notion of rough driver, that is an enriched version ofa time-dependent vector field, that is given by the additional datum of a time-dependentsecond order differential operator satisfying some algebraic and analytic conditions. Anotion of solution to a differential equation driven by a rough driver will be given, in theline of what was done in [Bai15] for rough differential equations, and the approximate flow-to-flow machinery will be seen to lead to a clean and simple well-posedness result for suchequations. Importantly, the Itô map, that associates to a rough driver the solution flow toits associated equation, is continuous. This continuity result is the key to deep results in thetheory of stochastic flows. We shall indeed prove that reasonable semimartingale velocityfields can be lifted to rough drivers under some mild boundedness and regularity conditions,and that the solution flow associated to the semimartingale rough driver coincides almostsurely with the solution flow to the Kunita-type Stratonovich differential equation drivenby the velocity field. In this sense, our theory of rough flows encompasses the theory ofstochastic flows. A Wong-Zakaï theorem will be proved for a general class of semimartingalevelocity fields and a sharp large deviation principle for Brownian flows will be proved willbe proved as a consequence of the continuity of the Itô map. No Wong-Zakaï result wasavailable so far for semimartingale stochastic flows.The setting of rough drivers and rough flows is presented in Section 1, together withthe approximate flow-to-flow machinery. This is the core of the deterministic machineryand everything that follows elaborates on this material, in a probabilistic setting. Someadditional material on random rough drivers is in particular given in Section 1.5, where weprovide some new variations around the Kolmogorov regularity theorem needed along theway and some sufficient conditions for a process to be bounded; this material may be ofindependent interest. We show in Section 2 that reasonable semimartingale velocity fieldscan be lifted to rough drivers under appropriate mild boundedness and regularity assump-tions and prove that the theory of semimartingale stochastic flows of maps is naturallyembedded in the theory of rough flows. As an illustration of the continuity of the Itô map,we prove in Section 2.5 a Wong-Zakaï theorem for stochastic flows of maps, and providein Section 3 a fresh look at large deviation theorems for Brownian stochastic flows. Weproved in the follow up work [BRS17] that a large class of Gaussian vector fields can belifted into rough drivers; this shows explicitly that the setting of rough drivers and roughflows goes beyond the semimartingale horizon. The point of view of rough flows presentedhere was also used in [BC17] to investigate the problem of stochastic turbulence.The size of this work is related to the fact that it is intended to be as self-containedas possible; no a priori knowledge of the theory of stochastic flows is required for itsunderstanding in particular. We have included as a consequence some material that iswell-known from experts in stochastic flows or large deviation theory for instance. A reader interested only in the machinery of rough flows will have a complete picture byreading sections 1.1 to 1.4 and Theorem 12 in Section 1.5. Notations.
We gather here for reference a number of notations that will be used throughoutthe text. • We shall exclusively use the letter E to denote a Banach space; we shall denote byL ( E ) the set of continuous linear maps from E to itself, and for M ∈ L ( E ) , we shallwrite | M | for its operator norm. In this setting, differentiability and regularitynotions are understood in the sense of Fréchet. • For functions x : [0 , T ] → E , we will use the notation x ts = x t − x s for the incrementsof x . • Whenever useful, vector fields are identified with the first order differential operatorthey define in a canonical way. • As is common, we shall use Einstein’s summation convention that a i b i := P i a i b i . • Last, recall that a flow on E is a family ( ϕ ts ) s t T of maps from E to itself suchthat ϕ tt = Id, for all t T , and ϕ tu ◦ ϕ us = ϕ ts , for all s u t T .The letter T , here and below, will always stand for a finite time horizon. Rough flows
We introduced in [Bai15] a simple machinery for con-structing flows on E from approximate flows that canbe understood as a generalization of Lyons’ workhorse [Lyo98] for constructing a roughpath from an almost rough path; this is the core tool for the construction of the roughintegral. Roughly speaking, the "Approximate flow-to-flow" machinery says that if weare given a family of maps ( µ ts ) s t T from E to itself, and if the maps µ are close todefining a flow, in the sense that µ tu ◦ µ us − µ ts is small in a quantitative way, for s u t with t − s small, then there exists a unique flow close to µ . In the rough paths setting,Lyons almost multiplicative functionals involve a family a = ( a ts ) s t T of elements of atensor algebra such that a tu a us − a ts has some given size whenever s u t with t − s small, with the product on the tensor space used here. Despite their similarity, Lyons’setting differs from the present setting in that multiplication in a tensor algebra satisfiesthe distributivity property ab − ac = a ( b − c ) , which obviously does not hold if a, b, c aremaps and the product stands for composition. This seemingly minor point makes a realdifference though, so it is fortunate that one can still get an analogue of Lyons’ theorem ina function space setting. This comes at a little price on the regularity of the set of maps µ that one can consider. As usual, for < r , we denote by C r the space of r -Hölderfunctions, with the understanding that they are Lipschitz continuous for r = 1 . Definition.
Let < r be given. A C r -approximate flow on E is a family (cid:0) µ ts (cid:1) s t T of (1 + ρ ) -Lipschitz maps from E to itself, for some < ρ , depending continuously on ( s, t ) in the topology of uniform convergence and enjoying the following two properties. • Perturbation of the identity –
There exists a constant α with < − ρ < α < , such that one has for any s t T , and any x ∈ E , the decomposition (1.1) D x µ ts = Id + A tsx + B tsx , for some L ( E ) -valued ρ -Lipschitz maps A ts with ρ -Lipschitz norm bounded above by c | t − s | α , and some L ( E ) -valued C bounded maps B ts , with C -norm bounded aboveby o t − s (1) . • C r -approximate flow property – There exists some positive constants c and a > , such that one has (1.2) (cid:13)(cid:13) µ tu ◦ µ us − µ ts (cid:13)(cid:13) C r c | t − s | a for all s u t T . Note that one requires a quantitative bound on A while we only require a qualitative in-formation on B , at the price of some more regularity for the latter. This fine decompositionof the differential of µ ts , as opposed to assuming only Dµ ts = Id + A ts , makes the settingmore flexible. The introduction of the notion of approximate flow is justified by the follow-ing result proved in [Bai15]. Given a partition π ts = { s = s < s < · · · < s n − < s n = t } of ( s, t ) ⊂ [0 , T ] , set µ π ts = µ s n s n − ◦ · · · ◦ µ s s . Theorem 1 (Constructing flows on E ) . A C r -approximate flow, with a < r , defines a uniqueflow ( ϕ ts ) s t T on E to which one can associate a positive constant δ such that the inequality (cid:13)(cid:13) ϕ ts − µ ts (cid:13)(cid:13) ∞ | t − s | a holds for all s t T with t − s δ ; this flow satisfies the inequality (1.3) (cid:13)(cid:13) ϕ ts − µ π ts (cid:13)(cid:13) ∞ c T (cid:12)(cid:12) π ts (cid:12)(cid:12) ar − for any partition π ts of any interval ( s, t ) ⊂ [0 , T ] , with mesh (cid:12)(cid:12) π ts (cid:12)(cid:12) δ . Moreover, the C r norm of the maps ϕ ts is uniformly bounded by a function of the constant c that appears in (1.2) , for all s t T . This statement generalises Gubinelli’ sewing lemma [Gub04], such as reshaped by Feyeland de la Pradelle in [FdLP06], to the non-commutative, non-associative setting of mapson E. (The non-commutative sewing lemma of Feyel, de la Pradelle and Mokobodski[FdLPM08] requires associativity and cannot be used in the present setting.)Theorem 1 is stated in [Bai15] for C -approximate flows; the proof given there worksverbatim for C r -approximate flows provided a < r ; a C map is then understood in thatsetting as a Lipschitz map. The crucial point in the above statement is the fact that if µ depends continuously in C r on some parameter then ϕ also happens to depend continuouslyon that parameter, in C , as a direct consequence of estimate (1.3). As made clear in[Bai15], Theorem 1 can be seen as the cornerstone of the theory of rough differentialequations, with the continuity of the Itô-Lyons solution map given as a consequence ofthe aforementioned continuity of ϕ on a parameter. We shall see in the present work thatTheorem 1 is all we need to get back and extend the core results of the theory of stochasticflows intensively developed in the 80’s and early 90’s. We shall need for that purpose tointroduce a notion of enriched velocity field that will somehow play the role in our settingof weak geometric Hölder p -rough paths, with p < , in rough paths theory. We shall thus pick a regularity exponent p < here, once and for all . Let us insisthere that like for the theory of rough paths, the technical shape of the theory of roughdrivers depends on that regularity exponent. Only two objects are needed in the definitionof a rough driver when p < ; for p < , we would need to introduce an additionalobject in the definition of a rough driver, that would thus consist of three objects; and so on. There is no real difficulty other than notational in giving a general theory, but asour applications of rough flows to the study of stochastic flows only require to develop thetheory in the case where p < , we stick to that setting and invite the reader to makeup herself her mind about what the general theory looks like. Let (cid:0) V ( · , t ) (cid:1) t T be a time-dependent vector field on E, with timeincrements V ts ( · ) := V ( · , t ) − V ( · , s ) . To get a hand on the definition of a weak geometric p -rough driver given below, think of V ts as given by the formula(1.4) V ts = V X ts , where V ( x ) ∈ L ( R ℓ , R d ) , for all x ∈ R d , and X = ( X, X ) is a p -rough path over R ℓ . Write V i for the image by V of the i th vector in the canonical basis of R ℓ . A solution path x • tothe rough differential equation dx t = V ( x t ) X ( dt ) can be characterized as a path satisfying some uniform Euler-Taylor expansion of the form f ( x t ) = f ( x s ) + X its (cid:0) V i f (cid:1) ( x s ) + X jkts (cid:0) V j V k f (cid:1) ( x s ) + O (cid:16) | t − s | p (cid:17) for all sufficiently regular real-valued functions f on R d . The present Section will makeit clear that the operators X its V i = V X ts and X jkts V j V k = ( DV ) V X ts are all we need inthis formula to run the theory, with no need to separate their space part, given by V and ( DV ) V , from their time part X ts . Definition.
Let p < , and p − < ρ be given. A weak geometric ( p, ρ ) - roughdriver is a family (cid:0) V ts (cid:1) s t T , with V ts := (cid:0) V ts , V ts (cid:1) , and V ts a second order differential operator, such that (i) the vector fields V ts are additive as functions of time V ts = V tu + V us for all s < u < t , and each V ts is of class C ρ on E, with sup s We shall freely talk about rough drivers ratherthan weak geometric ( p, ρ ) -rough drivers in the sequel. Definition. We define the (pseudo-)norm of V to be (1.5) k V k p,ρ := sup s We have (1.12) (cid:13)(cid:13)(cid:13) f ◦ µ ts − (cid:8) f + V ts f + V ts f (cid:9)(cid:13)(cid:13)(cid:13) ∞ c k f k C ρ | t − s | p , for any f ∈ C ρ and any s t T . The proof of this statement is straightforward and relies on the the following formula.For all x ∈ E and all f ∈ C , we have f (cid:0) µ ts ( x ) (cid:1) = f ( x ) + Z (cid:0) V ts f (cid:1) ( y u ) du + Z (cid:0) W ts f (cid:1) ( y u ) du = f ( x ) + (cid:0) V ts f (cid:1) ( x ) + (cid:0) V ts f (cid:1) ( x ) + ǫ fts ( x ) where ǫ fts ( x ) := Z Z u n(cid:0) V ts V ts f (cid:1) ( y r ) − (cid:0) V ts V ts f (cid:1) ( x ) o drdu + Z Z u (cid:0) W ts V ts f (cid:1) ( y r ) drdu + Z (cid:8)(cid:0) W ts f (cid:1) ( y u ) − (cid:0) W ts f (cid:1) ( x ) (cid:9) du. The inequality(1.13) (cid:13)(cid:13)(cid:13) ǫ fts (cid:13)(cid:13)(cid:13) C ρ c (cid:0) k V k (cid:1) k f k C ρ | t − s | p , justifies Proposition 2. Theorem 3. The family of maps (cid:0) µ ts (cid:1) s t T is a C ρ -approximate flow which depends con-tinuously on (cid:0) ( s, t ) , V (cid:1) in C topology. Proof – The family (cid:0) µ ts (cid:1) s t T satisfies the regularity assumptions (1.1) as a direct con-sequence of classical results on the dependence of solutions to ordinary differentialequations with respect to parameters, including the initial condition for the equation.These results also imply the continuous dependence of µ ts on (cid:0) ( s, t ) , V (cid:1) in C ρ topology.To show that the family µ defines a C ρ -approximate flow, write, for s u t T , µ tu (cid:0) µ us ( x ) (cid:1) = µ us ( x ) + V tu (cid:0) µ us ( x ) (cid:1) + (cid:0) V tu Id (cid:1)(cid:0) µ us ( x ) (cid:1) + ǫ Id tu (cid:0) µ us ( x ) (cid:1) = x + V us ( x ) + (cid:0) V us Id (cid:1) ( x ) + ǫ Id us ( x )+ V tu ( x ) + (cid:0) V us V tu (cid:1) ( x ) + (cid:0) V us V tu (cid:1) ( x ) + ǫ V tu us ( x )+ (cid:0) V tu Id (cid:1) ( x ) + (cid:16)(cid:0) V tu Id (cid:1)(cid:0) µ us ( x ) (cid:1) − (cid:0) V tu Id (cid:1) ( x ) (cid:17) + ǫ Id tu (cid:0) µ us ( x ) (cid:1) = µ ts ( x ) + n(cid:0) V us V tu (cid:1) ( x ) + (cid:16)(cid:0) V tu Id (cid:1)(cid:0) µ us ( x ) (cid:1) − (cid:0) V tu Id (cid:1) ( x ) (cid:17) + ǫ V tu us ( x )+ ǫ Id us ( x ) + ǫ Id tu (cid:0) µ us ( x ) (cid:1)o . The approximate flow property then follows from the regularity assumptions on V ts and V ts , and estimate (1.13). (cid:3) With the notations used in the definition of an approximate flow, the exponent a thatappears here in the approximate flow identity (1.2) is a = p . Definition 4. A flow (cid:0) ϕ ts (cid:1) s t T is said to solve the rough differential equation (1.14) dϕ = V ( ϕ ; dt ) if there exists a possibly V -dependent positive constant δ such that the inequality (1.15) (cid:13)(cid:13) ϕ ts − µ ts (cid:13)(cid:13) ∞ | t − s | p holds for all s t T with t − s δ . Flows solving a differential equation of the form (1.14) are called rough flows . If equation (1.14) is well-posed, the map which associates toa rough driver V the solution flow to equation (1.14) is called the Itô map . Following Cass and Weidner [CW17], one can equivalently take the Taylor expansionproperty ϕ ts = Id + V ts Id + V ts Id + O (cid:0) | t − s | p (cid:1) as a defining property of a solution flow to the rough differential equation (1.14). Thefollowing well-posedness result comes as a direct consequence of Theorem 1 and Theorem3. A family of maps is said to be uniformly C ρ is it has uniformly bounded C ρ -norm. Theorem 5. Assume ρ > p . Then the differential equation on flows dϕ = V ( ϕ ; dt ) has a unique solution flow; it takes values in the space of uniformly C ρ homeomorphisms of E , with uniformly C ρ inverses, and depends continuously on V in the topology of uniformconvergence. Proof – It follows from the proof of Theorem 3 that one can choose as a constant c ininequality (1.2) a multiple of k V k , so we have from Theorem 1 the estimate (cid:13)(cid:13) ϕ ts − µ π ts (cid:13)(cid:13) ∞ c (cid:0) k V k (cid:1) T (cid:12)(cid:12) π ts (cid:12)(cid:12) ρ p − , for any partition π ts of ( s, t ) ⊂ [0 , T ] with mesh (cid:12)(cid:12) π ts (cid:12)(cid:12) small enough, say no greaterthan δ . Note that the exponent ρ p − is positive. As these bounds are uniform in ( s, t ) , and for V in a bounded set of the space of rough drivers, and each µ π ts is acontinuous function of V , the flow ϕ depends continuously on (cid:0) ( s, t ) , V (cid:1) .To prove that ϕ is a homeomorphism, note that it follows from (1.8) that, for a b t , each µ ba is a diffeomorphism with inverse given by the time one map µ tt − a,t − b of the ordinary differential equation ˙ y u = − V ba ( y u ) − W ba ( y u ) = V tt − a,t − b ( y u ) + W tt − a,t − b ( y u ) , u , associated with the time reversed rough driver V t . As µ t has the same properties as µ , the maps (cid:0) µ π ts (cid:1) − = µ − s s ◦ · · · ◦ µ − s n s n − = µ ts n s n − ◦ · · · ◦ µ ts s converge uniformly to some continuous map ϕ − ts ,as the mesh of the partition π ts tendsto ; these limit maps ϕ − ts satisfy by construction ϕ ts ◦ ϕ − ts = Id.As Theorem 1 provides a uniform control of the C ρ norm of the maps ϕ ts , the samecontrol holds for their inverses since (cid:13)(cid:13) V t (cid:13)(cid:13) k V k . We propagate this control from theset (cid:8) ( s, t ) ∈ [0 , T ] ; s t, t − s δ (cid:9) to the whole of (cid:8) ( s, t ) ∈ [0 , T ] ; s t (cid:9) usingthe flow property of ϕ . (cid:3) Note that the solution flow to the rough differential equation dψ = V T ( ψ ; dt ) , driven by the time reversal of the rough driver V , from time T , provides the inverse flowof ϕ , in the sense that ϕ − ts = ψ T − s.T − t , for all s t T . Last, note that it is elementary to adapt the above results to add aglobally Lipschitz drift in the dynamics; the above results hold in that setting as well. Remark . The rough drivers introduced here are somewhat a dual version of similar objectsthat were introduced very recently in [BG18] by one of the authors and Gubinelli in the studyof the well-posedness of a general family of linear hyperbolic symmetric systems of equationsdriven by time-dependent vector fields that are only distributions in the time direction. Thelatter work deals with evolutions in function spaces and uses functional analytic tools in thesetting of controlled paths to make a first step towards a general theory of rough PDEs, inthe lines of the classical PDE approach based on duality, a priori estimates and compactnessresults. The present work does not overlap with the latter. How the story goes on. The entire technical core of the theory of rough flows iscontained in Section 1.1 and Section 1.3. The remainder of this work is dedicated to • showing how one can lift semimartingale velocity fields into rough drivers – Section1.5 and 2.3, • showing that stochastic and rough flows concide for semimartingale velocity fields– Section 2.4, • probing a Wong-Zakaï-type support theorem for semimartingale stochastic flows –Section 2.5, • proving some sharp Schilder and large deviation theorems for flows generated byGaussian rough drivers – Section 3.1.We emphasize here that we proved in the subsequent work [BRS17] that one can lift intorough drivers a whole class of Gaussian velocity fields, showing that the setting of roughflows goes beyond the setting of semimartingale calculus. With a view to identifying stochastic and rough flowsin Section 2.4, we prove here an elementary Itô formulaanalogue to Friz and Hairer’s Itô formula in [FH14]. As a matter of fact, Theorem 7 belowstates that any p -Hölder path in a Banach space satisfies an Itô formula, outside the settingof rough or controlled paths. To state and prove it recall Feyel and de la Pradelle sewinglemma [FdLP06], that can be seen as a precursor of the construction theorem for flows,Theorem 1. Given a partition π ts = { s = s < s < · · · < s n − < s n = t } of an interval [ s, t ] , and an E-valued -index map z = ( z ts ) s t T , set z π ts := z s n s n − + · · · + z s s . Theorem 6 ([FdLP06]) . Let (cid:0) z ts (cid:1) s t T be an E -valued -index continuous map to whichone can associate some positive constants c and a > such that (1.16) (cid:12)(cid:12)(cid:0) z tu + z us (cid:1) − z ts (cid:12)(cid:12) c | t − s | a holds for all s u t T . Then there exists a unique continuous function Z : [0 , T ] → R ,with increments Z ts := Z t − Z s ,to which one can associate a positive constant δ such that theinequality (cid:12)(cid:12) Z ts − z ts (cid:12)(cid:12) | t − s | a , holds for all s u t T , with t − s δ ; this map Z satisfies the inequality (cid:12)(cid:12) Z ts − z π ts (cid:12)(cid:12) c T (cid:12)(cid:12) π ts (cid:12)(cid:12) a − for any partition π ts of any interval [ s, t ] ⊂ [0 , T ] , with mesh (cid:12)(cid:12) π ts (cid:12)(cid:12) δ . It follows in particularthat Z depends continuously on any parameter in uniform topology if z does. A map z satisfying condition (1.16) is said to be almost-additive , and we write Z ts =: Z ts z du . We equip the tensor product space E ⊗ E with a compatible tensor norm that makes thenatural embedding L (cid:0) E , L ( E , R ) (cid:1) ⊂ L (cid:0) E ⊗ E , R (cid:1) continuous. Given such an choice, onecan identify the second differential of a C real-valued function on E to an element ofL (cid:0) E ⊗ E , R (cid:1) that is symmetric; this is what we do below. Theorem 7 (Itô formula) . Let F : [0 , T ] × E → R be a C -function of time with time derivative ∂ t F ( t, x ) bounded and continuous, uniformly in x ∈ E . Assume also that F is of class C in thesense of Fréchet as a function of its E -component, with derivatives F (1) , F (2) , F (3) and ∂ t F (1) ,bounded uniformly in time. Let ( x t ) s t T be p -Hölder E -valued map. Then the continuous -index map z ts := F (1)( s,x s ) ( x t − x s ) + 12 F (2)( s,x s ) ( x t − x s ) ⊗ is almost-additive, and we have (1.17) F (cid:0) t, x t (cid:1) = F (cid:0) s, x s (cid:1) + Z ts (cid:0) ∂ r F (cid:1) ( r, x r ) dr + Z ts z du , for any s t T . Proof – The proof is a straightforward application of Feyel-de la Pradelle’ sewing lemma,Theorem 6. Given s u t T , the algebraic identity z tu + z us = F (1)( s,x s ) ( x t − x s ) + (cid:16) F (1)( u,x u ) − F (1)( s,x s ) (cid:17) ( x t − x u )+ 12 F (2)( u,x u ) ( x t − x u ) ⊗ + 12 F (2)( s,x s ) ( x u − x s ) ⊗ , the regularity assumptions on F and the symmetric character of F (2)( s,x ) , for any x ∈ E,we have z tu + z us = F (1)( s,x s ) (cid:0) x t − x s (cid:1) + F (2)( s,x s ) ( x u − x s ) ⊗ ( x t − x u )+ O (cid:16) | t − s | p (cid:17) + O (cid:0) k x u − x s k (cid:1) (cid:13)(cid:13) x t − x u (cid:13)(cid:13) + 12 F (2)( u,x u ) ( x t − x u ) ⊗ + 12 F (2)( s,x s ) ( x u − x s ) ⊗ = z ts + O (cid:16) | t − s | p (cid:17) . Itô’s formula (1.17) follows by noting that we have for all n > F (cid:0) t, x t (cid:1) = n − X i =0 n F (cid:0) s i +1 , x s i +1 (cid:1) − F (cid:0) s i , x s i (cid:1)o = o n (1) + n − X i =0 ( s i +1 − s i ) ( ∂ s F ) (cid:0) s i , x s i (cid:1) + n − X i =0 n F (cid:0) s i , x s i +1 (cid:1) − F (cid:0) s i , x s i (cid:1)o , withF (cid:0) s i , x s i +1 (cid:1) − F (cid:0) s i , x s i (cid:1) = F (1)( s i ,x si ) (cid:0) x s i +1 − x s i (cid:1) + 12 F (2)( s i x si ) ( x s i +1 − x s i ) ⊗ + O (cid:16)(cid:12)(cid:12) x s i +1 − x s i (cid:12)(cid:12) (cid:17) = z s i +1 s i + O (cid:16) | s i +1 − s i | p (cid:17) . (cid:3) As an example, consider the solution flow ϕ to a rough differential equation on R d dϕ = V ( ϕ ; dt ) . Write ϕ t for ϕ t , and consider it as an element of the space E of continuous paths from [0 , T ] to C (cid:0) R d , R d (cid:1) , equipped with the norm of uniform convergence, with C (cid:0) R d , R d (cid:1) endowedwith a norm inducing uniform convergence on compact sets. It satisfies by its very definitionand Proposition 2 the Euler-Taylor expansion ϕ t = ϕ s + (cid:0) V ts Id (cid:1) ◦ ϕ s + (cid:0) V ts Id (cid:1) ◦ ϕ s + O (cid:0) | t − s | p (cid:1) so it is a p -Hölder path in that space. Now, given some points y , . . . , y k in R d and a C b real-valued function f on ( R d ) k , one can think of the function(1.18) F ( φ ) = f (cid:0) φ ( y ) , . . . , φ ( y k ) (cid:1) , for φ ∈ E, as a typical time-independent example of function satisfying the conditions ofTheorem 7. One then hasF (cid:0) ϕ s i +1 (cid:1) − F (cid:0) ϕ s i (cid:1) = f (cid:16) ϕ s i +1 s i (cid:0) ϕ s i ( y ) (cid:1) , . . . , ϕ s i +1 s i (cid:0) ϕ s i ( y k ) (cid:1)(cid:17) − f (cid:0) ϕ s i ( y ) , . . . , ϕ s i ( y k ) (cid:1) = k X m =1 (cid:16)(cid:0) V { m } s i +1 s i + V { m } s i +1 s i (cid:1) f (cid:17)(cid:0) ϕ s i ( y ) , . . . , ϕ s i ( y k ) (cid:1) + O c (cid:16) | s i +1 − s i | p (cid:17) , where the upper index { m } means that the operators act on the m th component of f . Theabove sum defines an almost-additive continuous function z fts , taken here at time (cid:0) s i +1 , s i (cid:1) ,so we have f (cid:0) ϕ t ( y ) , . . . , ϕ t ( y k ) (cid:1) = f (cid:0) ϕ s ( y ) , . . . , ϕ s ( y k ) (cid:1) + Z ts z fdu for all times s t T . We shall use below the theory of rough driversin a setting where the drivers are random. Likein the theory of rough paths, the primary object we are given is not the random roughdriver itself, or the random rough path, but rather a genuine random vector field, orrandom path, which needs to be enhanced in a first step into a random rough driver, orrandom rough path. This first, purely probabilistic, step can typically be done using someKolmogorov-type continuity arguments. We give in this Section some variations on thistheme that will be used to enhance vector field-valued martingales into rough drivers inSection 2.3; a reader interested only in these applications is advised to skip the technicaldetails and only have a look at Theorem 12; for the other readers, it is our hope that thissomewhat long section contains some material interesting in itself; it provides momentconditions under which one can get back uniform in time estimates on quantities of theform ( t − s ) − α k X ts k C a , such as required by the definition of a rough driver.The next Lemma gives sufficient conditions for a process defined on a possibly unboundeddomain to be bounded. Recall the equivalence of having Gaussian tails to square-rootgrowth of moments, cf. [FV10, Lemma A.17]. Lemma 8. Let ( E, d ) be a complete, separable metric space. Let D be an open subset of R d , X : D → ( E, d ) a continuous stochastic process, e ∈ E and κ > . Set D n := (cid:8) x ∈ D : n − | x | < n (cid:9) and N := { n ∈ N : D n = ∅} . Let ( a n ) n ∈ N be a sequence of non-negative real numbers and ( x n ) n ∈ N a sequence of elements in D such that x n ∈ D n for every n ∈ N .(i) Assume that there is a q ∈ [1 , ∞ ) and a γ ∈ ( dq , such that sup x,y ∈ D n (cid:13)(cid:13)(cid:13) d (cid:0) X ( x ) , X ( y ) (cid:1)(cid:13)(cid:13)(cid:13) L q κa n | x − y | γ and that (cid:13)(cid:13) d ( X ( x n ) , e ) (cid:13)(cid:13) L q κa n for every n ∈ N . Set ( b n ) := ( a n n γ ) and assume that k b k l q K < ∞ . Then there isa constant C = C ( q, γ ) such that (cid:13)(cid:13) sup x ∈ D d ( X ( x ) , e ) (cid:13)(cid:13) L q CKκ. (ii) Let γ ∈ (0 , and assume that for every q > there is a c q such that for every n ∈ N , sup x,y ∈ D n (cid:13)(cid:13)(cid:13) d (cid:0) X ( x ) , X ( y ) (cid:1)(cid:13)(cid:13)(cid:13) L q κc q a n | x − y | γ and that (cid:13)(cid:13) d ( X ( x n ) , e ) (cid:13)(cid:13) L q κc q a n where c q = O ( √ q ) when q → ∞ . Assume that a n = O (cid:16) n − γ (cid:0) n ) (cid:1) − (cid:17) . Thenfor every q > there is some constant C = C ( q, γ ) such that (cid:13)(cid:13)(cid:13) sup x ∈ D d (cid:0) X ( x ) , e (cid:1)(cid:13)(cid:13)(cid:13) L q Cκ with C = O ( √ q ) when q → ∞ . In particular, the random variable sup x ∈ D d ( X ( x ) , e ) has Gaussian tails. Proof – Without loss of generality, one may assume κ = 1 , otherwise we consider themetric ˜ d = d/κ instead, and N = N – otherwise we add small, disjoint balls to D anddefine X to be constant and equal to e on these balls. We first prove claim (i).Let α > dq and set p ( u ) = u α + dq . By the Garsia-Rodemich-Rumsey Lemma (cf. e.g.[Sch09, Lemma 2.4 (i)]), for every x, y ∈ D n , d (cid:0) X ( x ) , X ( y ) (cid:1) | x − y | α − dq CV q n where V n = Z D n × D n (cid:12)(cid:12)(cid:12) d (cid:0) X ( u ) , X ( v ) (cid:1)(cid:12)(cid:12)(cid:12) q | u − v | αq + d du dv. Thus, by a change of variables, E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup x,y ∈ D n d (cid:0) X ( x ) , X ( y ) (cid:1) | x − y | α − dq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q C q a qn Z D n × D n | u − v | ( γ − α ) q − d du dv C q a qn n d +( γ − α ) q Z (0 , | u − v | ( γ − α ) q − d du dv. Let β ∈ (0 , γ − dq ) and set α = dq + β < γ . Then the integral is finite, and we haveshown that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup x,y ∈ D n d (cid:0) X ( x ) , X ( y ) (cid:1) | x − y | β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q Ca n n ( γ − β ) . By the triangle inequality, (cid:13)(cid:13)(cid:13)(cid:13) sup x ∈ D n d (cid:0) X ( x ) , e (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) L q Ca n ( n γ + 1) Cb n . Thus we obtain E (cid:18) sup x ∈ D d (cid:0) X ( x ) , e (cid:1) q (cid:19) = E (cid:18) sup n sup x ∈ D n d (cid:0) X ( x ) , e (cid:1) q (cid:19) ∞ X n =1 E (cid:18) sup x ∈ D n d (cid:0) X ( x ) , e (cid:1) q (cid:19) q C q ∞ X n =1 b qn < ∞ and claim (i) is shown.Now we prove claim (ii). Note that the constant in the Garsia-Rodemich-RumseyLemma may be chosen non-increasing in q . Therefore, we can argue similarly asbefore to see that for every q > and n ∈ N , (cid:13)(cid:13)(cid:13)(cid:13) sup x ∈ D n d (cid:0) X ( x ) , e (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) L q C q b n where C q = O ( √ q ) . This shows that the random variable has Gaussian tails, i.e. thereis some constant C such that for every n ∈ NP (cid:18) sup x ∈ D n d (cid:0) X ( x ) , e (cid:1) > t (cid:19) C exp (cid:18) − t Cb n (cid:19) for every t > . Hence P (cid:18) sup x ∈ D d (cid:0) X ( x ) , e (cid:1) > t (cid:19) C ∞ X n =1 exp (cid:18) − t Cb n (cid:19) C ∞ X n =1 exp (cid:18) − t C (1 + log( n )) (cid:19) C exp (cid:18) − t C (cid:19) ∞ X n =1 n − t C and the sum is finite for t large enough. This proves that sup x ∈ D d ( X ( x ) , e ) hasGaussian tails. (cid:3) Corollary 9. Let D be an open subset of R d , ( E , k · k ) a separable Banach space, X : D → E a continuous stochastic process and q > .(i) Assume that there is a constant κ > and γ ∈ (0 , such that for every x, y ∈ D with < | x − y | , (cid:13)(cid:13) X ( x ) − X ( y ) (cid:13)(cid:13) L q κ | x − y | γ (1.19) and that there is an η ∈ (0 , ∞ ) such that for every x ∈ D , k X ( x ) k L q κ | x | η (1.20) with q sufficiently large satisfying q > η + d (cid:18) η + 1 γ (cid:19) . Then the random variable sup x ∈ D k X ( x ) k is almost surely finite. Moreover, there is aconstant C = C ( γ, q, η ) such that (cid:13)(cid:13)(cid:13)(cid:13) sup x ∈ D (cid:13)(cid:13) X ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q Cκ. (1.21) (ii) Assume that (1.19) and (1.20) hold for every q > with κ = κ ( q ) √ q ˆ κ and some η ∈ (0 , ∞ ) .Then sup x ∈ D k X ( x ) k has Gaussian tails and there is a constant C = C ( γ, η ) suchthat (cid:13)(cid:13)(cid:13)(cid:13) sup x ∈ D (cid:13)(cid:13) X ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q C √ q ˆ κ for every q > . Proof – We start with proving (i). Let x, y ∈ D such that | x − y | > . Then, by (1.20), (cid:13)(cid:13) X ( x ) − X ( y ) (cid:13)(cid:13) L q (cid:13)(cid:13) X ( x ) (cid:13)(cid:13) L q + (cid:13)(cid:13) X ( y ) (cid:13)(cid:13) L q κ κ | x − y | γ which shows that (cid:13)(cid:13) X ( x ) − X ( y ) (cid:13)(cid:13) L q κ | x − y | γ (1.22)holds for every x, y ∈ D . Now let x, y ∈ D such that n − | x | , | y | < n . Interpolatingbetween the inequality (cid:13)(cid:13) X ( x ) − X ( y ) (cid:13)(cid:13) L q (cid:13)(cid:13) X ( x ) (cid:13)(cid:13) L q + (cid:13)(cid:13) X ( y ) (cid:13)(cid:13) L q κ n − η and inequality (1.22), we see that for every λ ∈ [0 , , (cid:13)(cid:13) X ( x ) − X ( y ) (cid:13)(cid:13) L q Cκn − (1 − λ ) η | x − y | γλ and (cid:13)(cid:13) X ( x ) (cid:13)(cid:13) L q Cκn − (1 − λ ) η for every x, y ∈ D n . Set γ ′ := γλ and a n := n − (1 − λ ) η . In order to obtain ( a n n γ ′ ) ∈ ℓ q ( N ) , we must have q ( λγ − (1 − λ ) η ) < − which is equivalent to q < η − λ ( η + γ ) . The condition γ ′ > dq is equivalent to q < λγd . Choosing λ ∗ = ηγ/d + η + γ ∈ (0 , , we have η − λ ∗ ( η + γ ) = λ ∗ γd = γηγ + d ( γ + η ) which is indeed smaller than q by assumption. Hence we may apply Lemma 8 toconlude (i). The claim (ii) follows by applying Lemma 8 (ii). (cid:3) Example. Consider the Gaussian process X : (0 , ∞ ) → R where X t = B t t α , for a standardBrownian motion B , and α ∈ ( , . Then, if t > , k X t k L q . √ q k X t k L . √ q t α − and for s < t , k X t − X s k L = k B t (1 + s α ) − B s (1 + t α ) k L (1 + s α )(1 + t α ) k B t − B s k L t α + k B s k L | t α − s α | (1 + s α )(1 + t α ) | t − s | t α + 2 s t α − (1 + s α )(1 + t α ) | t − s | | t − s | and (cid:13)(cid:13) X t − X s (cid:13)(cid:13) L q . √ q (cid:13)(cid:13) X t − X s (cid:13)(cid:13) L . Applying part (ii) in Corollary 9 shows that the random variable sup t ∈ (0 , ∞ ) (cid:12)(cid:12) X t (cid:12)(cid:12) is finite and has Gaussian tails. Note that this is sharp in the sense that the law of theiterated logarithm for a Brownian motion implies that it is not possible to choose α = . Next, we apply the same ideas to give conditions for Hölder continuity. Lemma 10. Let ( E, d ) be a complete separable metric space, D an open subset of R d , X : D → ( E, d ) a continuous stochastic process and κ > . Set D n := n x ∈ D : n − | x | < n o and N := { n ∈ N : D n = ∅} . Let ( a n ) n ∈ N be a sequence of non-negative real numbers.(i) Assume that there is a q > and a γ ∈ ( dq , such that for every n ∈ N and every x, y ∈ D n with < | x − y | , k d ( X ( x ) , X ( y )) k L q κa n | x − y | γ . Let β ∈ (cid:0) , γ − dq (cid:1) . Define the sequence ( b n ) := (cid:0) a n n γ − β (cid:1) , and assume that k b k l q K < ∞ . Then there is a constant C = C ( q, γ ) such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup x,y ∈ D < | x − y | d (cid:0) X ( x ) , X ( y ) (cid:1) | x − y | β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q CKκ. (ii) Assume that there is some γ ∈ (0 , and that for every q > there is a c q such thatfor every n ∈ N and every x, y ∈ D n with < | x − y | , (cid:13)(cid:13)(cid:13) d (cid:0) X ( x ) , X ( y ) (cid:1)(cid:13)(cid:13)(cid:13) L q κc q a n | x − y | γ where c q = O ( √ q ) when q → ∞ . Let β ∈ (0 , γ ) and assume that a n = O (cid:16) n − ( γ − β ) (1+log( n )) − (cid:17) . Then for every q > there is some constant C = C ( q, γ ) such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup x,y ∈ D < | x − y | d (cid:0) X ( x ) , X ( y ) (cid:1) | x − y | β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q Cκ with C = O ( √ q ) when q → ∞ . In particular, the random variable sup x,y ∈ D < | x − y | d (cid:0) X ( x ) , X ( y ) (cid:1) | x − y | β has Gaussian tails. Proof – Without loss of generality, one can choose κ = 1 and N = N . For n ∈ N , set ˜ D n := (cid:8) D n ∪ D n +1 ∪ D n +2 (cid:9) . We first prove (i). Fix some n ∈ N and some k ∈ N . Let α > dq and define p k ( s ) = ( s α + dq if s ∈ [0 , αq + d + k ( s − q if s > . Fix x, y ∈ ˜ D n with < | x − y | . From the Garsia-Rodemich-Rumsey Lemma, d (cid:0) X ( x ) , X ( y ) (cid:1) CV q n,k | x − y | α − dq where V n,k = Z ˜ D n × ˜ D n (cid:12)(cid:12)(cid:12) d (cid:0) X ( u ) , X ( v ) (cid:1)(cid:12)(cid:12)(cid:12) q p k (cid:0) | u − v | (cid:1) q du dv. Thus E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup x,y ∈ ˜ Dn < | x − y | d (cid:0) X ( x ) , X ( y ) (cid:1) | x − y | α − dq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q C q Z ˜ D n × ˜ D n E h d (cid:0) X ( u ) , X ( v ) (cid:1) q i p k ( | u − v | ) q du dv C q X l =0 , , Z D n + l × D n + l E h d (cid:0) X ( u ) , X ( v ) (cid:1) q i p k ( | u − v | ) q du dv. For every m ∈ N , we have Z D m × D m E h d (cid:0) X ( u ) , X ( v ) (cid:1) q i p k ( | u − v | ) q du dv a qm Z ( D m × D m ) ∩{| u − v | } | u − v | ( γ − α ) q − d du dv + Z ( D m × D m ) ∩{| u − v | > } E h d (cid:0) X ( u ) , X ( v ) (cid:1) q i αq + d + k ( | u − v | − du dv Moreover, by a change of variables, Z ( D m × D m ) ∩{| u − v | } | u − v | ( γ − α ) q − d du dv Z D m × D m | u − v | ( γ − α ) q − d du dv = m d +( γ − α ) q Z (0 , | u − v | ( γ − α ) q − d du dv. Set α = dq + β < γ . Then this integral is finite, and sending k → ∞ shows that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup x,y ∈ ˜ Dn < | x − y | d (cid:0) X ( x ) , X ( y ) (cid:1) | x − y | β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q C (cid:16) a n n ( γ − β ) + a n +1 ( n + 1) ( γ − β ) + a n +2 ( n + 2) ( γ − β ) (cid:17) = C (cid:0) b n + b n +1 + b n +2 (cid:1) . Now take x, y ∈ D with < | x − y | and assume that d (cid:0) X ( x ) , X ( y ) (cid:1) | x − y | β > t. Then there is an n ∈ N such that x ∈ D n and since | x − y | , y ∈ (cid:8) D n − ∪ D n ∪ D n +1 (cid:9) ,where we set D := D . Thus we have shown that for every t > , sup x,y ∈ D < | x − y | d (cid:0) X ( x ) , X ( y ) (cid:1) | x − y | β > t ⊆ [ n ∈ N sup x,y ∈ ˜ Dn < | x − y | | d (cid:0) X ( x ) , X ( y ) (cid:1) | x − y | β > t and therefore E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup x,y ∈ D < | x − y | d (cid:0) X ( x ) , X ( y ) (cid:1) | x − y | β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ∞ X n =1 E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup x,y ∈ ˜ Dn < | x − y | d (cid:0) X ( x ) , X ( y ) (cid:1) | x − y | β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q C q K q . Now we prove (ii). Note that the constant in the Garsia-Rodemich-Rumsey Lemmamay be chosen non-increasing in q . Therefore, we can argue similarly as before to seethat for every q > and n ∈ N , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup x,y ∈ ˜ Dn < | x − y | d (cid:0) X ( x ) , X ( y ) (cid:1) | x − y | β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q C q (cid:0) b n + b n +1 + b n +2 (cid:1) where C q = O ( √ q ) . This shows that the random variable has Gaussian tails, i.e. thereis some constant C such that for every n ∈ NP sup x,y ∈ ˜ Dn < | x − y | d (cid:0) X ( x ) , X ( y ) (cid:1) | x − y | β > t C exp (cid:18) − t C ( b n + b n +1 + b n +2 ) (cid:19) for every t > . Hence P sup x,y ∈ D < | x − y | d (cid:0) X ( x ) , X ( y ) (cid:1) | x − y | β > t C ∞ X n =1 exp (cid:18) − t C ( b n + b n +1 + b n +2 ) (cid:19) C ∞ X n =1 exp (cid:18) − t C (1 + log( n )) (cid:19) C exp (cid:18) − t C (cid:19) ∞ X n =1 n − t /C and the sum is finite for t large enough. This proves that sup x,y ∈ D < | x − y | d (cid:0) X ( x ) , X ( y ) (cid:1) | x − y | β has Gaussian tails. (cid:3) Example. Let X : (0 , ∞ ) → R be the Gaussian process defined as X t = B t p t log(1 + t ) ,B being a standard Brownian motion. Then k X t − X s k L | t − s | p t log(1 + t ) + p t log(1 + t ) − p s log(1 + s ) p log(1 + s ) p s log(1 + s ) . By the mean value theorem, t log(1 + t ) − s log(1 + s ) (log(1 + t ) + 1)( t − s ) and therefore p t log(1 + t ) − p s log(1 + s ) p t log(1 + t ) − s log(1 + s ) p (log(1 + t ) + 1) | t − s | . If ( n − s t n , we have for any q > k X t − X s k L q . √ q k X t − X s k L . √ qa n | t − s | with a n = O (cid:16) n − (1 + log( n )) − (cid:17) . Part (ii) of Lemma 10 shows that for any β ∈ (cid:0) , (cid:1) , therandom variable sup s,t ∈ (0 , ∞ ) < | t − s | (cid:12)(cid:12) X t − X s (cid:12)(cid:12) | t − s | β is finite and has Gaussian tails. Corollary 11. Let D be an open subset of R d , and (cid:0) E, k · k (cid:1) be a separable Banach space.Let X : D → E be a continuous stochastic process and q > .(i) Assume that there are constants κ > , γ ∈ (0 , and β ∈ (0 , γ ) such that for every x, y ∈ D with < | x − y | , (cid:13)(cid:13) X ( x ) − X ( y ) (cid:13)(cid:13) L q κ | x − y | γ (1.23) and that there is an η ∈ (0 , ∞ ) such that for every x ∈ D , (cid:13)(cid:13) X ( x ) (cid:13)(cid:13) L q κ | x | η (1.24) where q > satisfies q > γη ( γ − β ) + d (cid:18) γη ( γ − β ) + 1 γ − β (cid:19) . Then the random variable sup x,y ∈ D < | x − y | (cid:13)(cid:13) X ( x ) − X ( y ) (cid:13)(cid:13) | x − y | β is almost surely finite. Moreover, there is a constant C = C ( γ, η, β, q ) such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup x,y ∈ D < | x − y | (cid:13)(cid:13) X ( x ) − X ( y ) (cid:13)(cid:13) | x − y | β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q Cκ. (1.25) (ii) Assume that (1.23) and (1.24) hold for every q > with κ = κ ( q ) √ q ˆ κ and some η ∈ (0 , ∞ ) .Then for every β ∈ (0 , γ ) , sup x,y ∈ D < | x − y | (cid:13)(cid:13) X ( x ) − X ( y ) (cid:13)(cid:13) | x − y | β has Gaussian tails, and there is a constant C = C ( γ, η, β ) such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup x,y ∈ D < | x − y | (cid:13)(cid:13) X ( x ) − X ( y ) (cid:13)(cid:13) | x − y | β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q C √ q ˆ κ holds for every q > . Proof – The proof is similar to the proof of Corollary 9, using Lemma 10 above. We leavethe details to the reader. (cid:3) If D is an open subset of R d , and (cid:0) E, k · k (cid:1) is a normed space, f : D → E a function and ρ ∈ (0 , , we define k f k ∼C ρ := max x ∈ D (cid:13)(cid:13) f ( x ) (cid:13)(cid:13) , sup x,y ∈ D < | x − y | (cid:13)(cid:13) f ( x ) − f ( y ) (cid:13)(cid:13) | x − y | ρ . Let f, g : D → R m . Then we define the function ( f ⊙ g ) : D → R m × m by setting ( f ⊙ g ) ij ( x ) = f i ( x ) g j ( x ) . Note that k · k ∼C ρ is equivalent to k · k C ρ and that we have k f ⊙ g k ∼C ρ k f k ∼C ρ k g k ∼C ρ (1.26)provided we equip R m and R m × m with the sup norm.In the following, we will consider stochastic processes V : D × [0 , T ] → R m and W : D ×{ s t T } → R m × m for which we assume that for every s < u < t ∈ [0 , T ] and every x ∈ D , W ts ( x ) − W us ( x ) − W tu ( x ) = V us ( x ) ⊙ ˜ V tu ( x ) (1.27)holds almost surely. The next theorem is the main result of this section. Theorem 12 (Kolmogorov criterion for rough drivers) . Let D be an open subset of R d , κ > and γ , γ ∈ (0 , .(i) Let V : D × I → R m be a stochastic process and q > . Assume that for every x, y ∈ D with < | x − y | and s < t ∈ I , (cid:13)(cid:13) V ts ( x ) − V ts ( y ) (cid:13)(cid:13) L q κ | t − s | γ | x − y | γ (1.28) and that there is an η ∈ (0 , ∞ ) such and that for every x ∈ D and s < t ∈ I , (cid:13)(cid:13) V ts ( x ) (cid:13)(cid:13) L q κ | t − s | γ | x | η . (1.29) Let α ∈ (0 , γ ) , β ∈ (0 , γ ) and assume that q > max (cid:26) γ η ( γ − β ) + d (cid:18) γ η ( γ − β ) + 1 γ − β (cid:19) , γ − α (cid:27) . (1.30) Then there is a continuous modification of the process V . Moreover, there is a constant C = C ( γ , γ , α, β, η, d, T, q ) such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup s Without loss of generality we may assume κ = 1 , otherwise we can replace V and W by V /κ resp. W/κ . Furthermore, we will prove the result for the k · k ∼C β norm,claimed results follow by equivalence of norms.We start with proving (i). Fix s < t . Using (1.28) and the classical Kolmogorovtheorem [Kal02, Theorem 3.23], there is a continuous modification of the process x V ts ( x ) on D . The estimates (1.28) and (1.29) and Corollary 11 imply that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup x,y ∈ D < | x − y | (cid:12)(cid:12) V ts ( x ) − V ts ( y ) (cid:12)(cid:12) | x − y | β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q C | t − s | γ and Corollary 9 gives (cid:13)(cid:13)(cid:13)(cid:13) sup x ∈ D (cid:12)(cid:12) V ts ( x ) (cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13) L q C | t − s | γ . Note in particular that the constant on the right hand side of both equations is inde-pendent of s and t . We can repeat this procedure for every s < t and obtain a process t V t which, for every t ∈ [0 , T ] , takes values in C βb almost surely, and for which (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V t − V s (cid:13)(cid:13) ∼C β (cid:13)(cid:13)(cid:13) L q C | t − s | γ (1.35)holds for every s < t . Applying again the Kolmogorov theorem for Banach spacevalued processes gives the claim.We proceed with (ii). As in (i), for every s < t there are modifications of the process x W ts ( x ) such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) W ts (cid:13)(cid:13) ∼C β (cid:13)(cid:13)(cid:13) L q C | t − s | γ . Using the algebraic relation (1.27), the estimate (1.35) for V and the compatibility ofthe k · k ∼C β norms given in (1.26), we can mimic the proof of the Kolmogorov criterionfor rough paths ([FH14, Theorem 3.1]) to conlude.Assertion (iii) follows similarly by using part (ii) in the Corollaries 11 and 9. (cid:3) Finally, we give a Kolmogorov criterion for the distance between rough drivers, whoseproof is very similar to the proof of Theorem 12, using the Kolmogorov criterion for roughpath distance [FH14, Theorem 3.3]; we leave it to the reader. Theorem 13 (Kolmogorov criterion for rough driver distance) . Let κ > , γ , γ ∈ (0 , and ( V, W ) , ( ˆ V , ˆ W ) processes as in Theorem 12. Set ∆ V := V − ˆ V and ∆ W := W − ˆ W .(i) Assume that ( V, W ) and ( ˆ V , ˆ W ) satisfy the same moment conditions as in Theorem12 with q sufficiently large as in (1.30) . Moreover, assume that there is an ε > suchthat (cid:13)(cid:13) ∆ V ts ( x ) − ∆ V ts ( y ) (cid:13)(cid:13) L q εκ | t − s | γ | x − y | γ (1.36) and (cid:13)(cid:13) ∆ W ts ( x ) − ∆ W ts ( y ) (cid:13)(cid:13) L q εκ | t − s | γ | x − y | γ (1.37) for every x, y ∈ D such that < | x − y | and every s < t ∈ [0 , T ] and that (cid:13)(cid:13) ∆ V ts ( x ) (cid:13)(cid:13) L q ε κ | t − s | γ | x | η (1.38) and (cid:13)(cid:13) ∆ W ts ( x ) (cid:13)(cid:13) L q ε κ | t − s | γ | x | η (1.39) hold for every x ∈ D and every s < t ∈ [0 , T ] . Then there are continuous modi-fications of the processes ( V, W ) and ( ˆ V , ˆ W ) . Moreover, there is a constant C = C ( γ , γ , α, β, η, d, T, q ) such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup s The theory of stochastic flows grew out of the pioneering works of the Russian school[BF61, IA72] on the dependence of solutions to stochastic differential equations with re-spect to parameters and the proof by Bismut [Bis81] and Kunita [Kun81c] that stochasticdifferential equations generate continuous flows of diffeomorphisms under proper regularityconditions on the driving vector fields. The Brownian character of these random flows, thatis the fact that they are continuous with stationary and independent increments, was in-herited from the Brownian character of their driving noise. The next natural step consistedin the study of Brownian flows for themselves. After the works of Harris [Har81], Baxen-dale [Bax80] and Le Jan [LJ82], they appeared to be generated by stochastic differentialequations driven by infinitely many Brownian motions, or better, to be in one-to-one corre-spondence with vector field-valued Brownian motions. A probabilistic integration theory ofsuch random time-varying velocity fields was developed to establish that correspondence,and it was extended by Le Jan and Watanabe [LJW84] to a large class of continuous semi-martingale flows and continuous semimartingale velocity fields. Kunita [Kun81a, Kun86]studied the problem of convergence of stochastic flows, with applications to averaging andhomogenization results, and promoted the use of stochastic flows to implement a versionof the characteristic method in the setting of first and second order stochastic partialdifferential equations, notably those coming from the nonlinear filtering theory.We shall show in this Section that the theory of semimartingale stochastic flows can beembedded into the theory of rough flows developed in Section 1. We review in Section2.2 the basics of the theory of stochastic flows and show in Section 2.3 that sufficientlyregular (semi)martingale velocity fields can be lifted to rough drivers; this is done us-ing our Kolmogorov-type criterion for rough drivers, Theorem 12. The identification of(semi)martingale flows generated by (semi)martingale velocity fields to rough flows asso-ciated with the corresponding rough driver is done through the Itô formula, on which onecan read the local characteristics of a semimartingale flow. The study of stochastic flows classically requires theintroduction of a number of function spaces, that werecall here.Let E and F be Banach spaces. The derivative of a function f from E to F is understoodin the Fréchet sense. We shall equip tensor products of Banach spaces with a compatible tensor norm which makes the canonical embedding L (cid:0) E, L ( E, F ) (cid:1) ֒ → L (cid:0) E ⊗ E, F (cid:1) continuous. The n -th derivative of f can be seen as a function D n f : E → L (cid:0) E ⊗ n , F (cid:1) . For n ∈ N and ρ ∈ (0 , , we define k f k n + ρ := k f k C n + ρ := n X i =0 sup x ∈ E (cid:13)(cid:13) D i f ( x ) (cid:13)(cid:13) + sup < k x − y k (cid:13)(cid:13) D n f ( x ) − D n f ( y ) (cid:13)(cid:13) k x − y k ρ . We define C n,ρb ( E, F ) to be the space of n -times continuously differentiable functions f : E → F such that k f k C n + ρ < ∞ .Next, we consider the finite dimensional case. Let be D be a domain of R d , A ⊆ D asubset, n ∈ N and ρ ∈ (0 , . For a function f : D → R k , set k f k n + ρ ; A := X | α | n sup x ∈ A (cid:12)(cid:12) D α f ( x ) (cid:12)(cid:12) + X | α | = n sup x,y ∈ A < | x − y | (cid:12)(cid:12) D α f ( x ) − D α f ( y ) (cid:12)(cid:12) | x − y | ρ . We also set k f k n + ρ := k f k n + ρ ; D . Note that this is consistent with the notation above when D = R d . Let C n,ρ ( D, R k ) be the space of n -times continuously differentiable functions f : D → R k such that k f k n + ρ,K < ∞ for every compact subset K ⊂ D . Note thatalthough the (semi-)norms we defined here differ slightly from those used by Kunita in hisbook [Kun90], they are actually equivalent on compact sets, hence the spaces coincide. Wealso define C n,ρb ( D, R k ) := n f ∈ C n,ρ ( D, R k ) : k f k n + ρ < ∞ o . For a function g : D × D → R k , we similarly define k g k ∧ n + ρ ; A := X | α | n sup x,y ∈ A (cid:12)(cid:12) D αx D αy g ( x, y ) (cid:12)(cid:12) + X | α | = n sup x,y,x ′ ,y ′∈ A < | x − x ′ | , | y − y ′ | (cid:12)(cid:12)(cid:12) D αx D αy g ( x, y ) − D αx D αy g ( x ′ , y ) − D αx D αy g ( x, y ′ ) + D αx D αy g ( x ′ , y ′ ) (cid:12)(cid:12)(cid:12) | x − x ′ | ρ | y − y ′ | ρ . As above, set k g k ∧ n + ρ := k g k ∧ n + ρ ; D . We denote by b C n,ρ ( D × D, R k ) the space of functions g : D × D → R k which are n -times continuously differentiable with respect to each x and y and for which k g k ∧ n + ρ ; K < ∞ for every compact subset K ⊂ D . Set b C n,ρb ( D × D, R k ) := n g ∈ b C n,ρ ( D × D, R k ) : k g k ∧ n + ρ < ∞ o . We will sometimes use the shorter notation C n,ρ , C n,ρb , b C n,ρ resp. b C n,ρb when domain andcodomain of the function spaces are clear from the context. We describe in this Section the basics of the theory ofsemimartingale stochastic flows, and refer the readerto [LJW84] or [Kun90] for a complete account; we refer to Kunita’s book for preciseregularity and growth assumptions on the different objects involved. Readers familiar withthis material can go directly to Section 2.3.Let (cid:0) Ω , F , ( F t ) t T , P (cid:1) be a filtered probability space; denote by Diff , resp. F , thecomplete separable metric spaces of C k diffeomorphisms of R d , resp. C k vector fields on R d , for some integer k > . Definition. A Diff -valued continuous (cid:0) F t (cid:1) t T -adapted random process ( φ t ) t T is called a Diff -valued semimartingale stochastic flow of maps if the real-valued processes f (cid:0) φ • ( x ) (cid:1) are real-valued (cid:0) F t (cid:1) t T -semimartingales for all x ∈ R d , and all f ∈ C ∞ c ( R d ) . Such a Diff -valued semimartingale is said to be regular if for every x, y ∈ R d , and f, g ∈ C ∞ c ( R d ) ,the bounded variation part of f (cid:0) φ • ( x ) (cid:1) and the bracket (cid:10) f (cid:0) φ • ( x ) (cid:1) , g (cid:0) φ • ( y ) (cid:1)(cid:11) are absolutelycontinuous with respect to Lebesgue measure dt . Their densities w ft ( x ) and { f, g } t ( x, y ) can be chosen to be jointly measurable andcontinuous in f, g in the C -norm [LJW84]. Set (cid:0) L t f (cid:1) ( x ) := w ft (cid:0) φ − t ( x ) (cid:1) , h f, g i t ( x, y ) := { f, g } t (cid:0) φ − t ( x ) , φ − t ( y ) (cid:1) , so that the processes M ft ( x ) := f (cid:0) φ t ( x ) (cid:1) − f (cid:0) φ ( x ) (cid:1) − Z t (cid:0) L s f (cid:1)(cid:0) φ s ( x ) (cid:1) ds, x ∈ R d , f ∈ C ∞ c ( R d ) are continuous (cid:0) F t (cid:1) t T -local martingales with bracket (cid:10) M f ( x ) , M g ( y ) (cid:11) t = Z t h f, g i t (cid:0) φ s ( x ) , φ s ( y ) (cid:1) ds. We have (cid:0) L t f (cid:1) ( x ) = lim h ↓ E " f (cid:0) φ t + h,t ( x ) (cid:1) − f ( x ) h (cid:12)(cid:12)(cid:12)(cid:12) F t and h f, g i ( x, y ) = lim h ↓ h E h(cid:8) f (cid:0) φ t + h,t ( x ) (cid:1) − f ( x ) (cid:9)(cid:8) g (cid:0) φ t + h,t ( y ) (cid:1) − g ( y ) (cid:9)(cid:12)(cid:12)(cid:12) F t i , with limits in L whenever they exist. Under proper regularity conditions [LJW84], theoperators h f, g i s ( x, y ) can be seen to be random differential operators of the form h f, g i s ( x, y ) = A ijs ( x, y ) ∂ x i y j , for some process A s ( x, y ) with values in the space Symm ( d ) of symmetric d × d matrices.The operators L s can moreover be expressed in terms of A s and its differential with respectto the space variables, so that the data of the processes A s and L s is equivalent to the dataof the process A s and an F -valued process b • . The family of random operators h· , ·i t and the drift b t are called the local characteristics of the Diff -valued semimartingale φ • . Asan example, for the semimartingale flow generated by a stochastic differential equation ofthe form dx t = V i ( x t ) ◦ dB it , driven by an ℓ -dimensional Brownian motion, we have L t f = 12 ℓ X i =1 V i f and h f, g i s ( x, y ) = (cid:0) V i f (cid:1) ( x ) (cid:0) V j g (cid:1) ( y ) , and the drift b s in the local characteristic is given here by the time-independent vectorfield b s ( x ) = 12 (cid:0) V i V i (cid:1) ( x ) = 12 (cid:0) D x V i (cid:1) V i ( x ) . The infinitesimal counterpart of a Diff -valued semimartingale is given by the followingnotion. Definition. A semimartingale velocity field is an F -valued process ( V t ) t T such that theprocesses (cid:0) V • f (cid:1) ( x ) are real-valued semimartingale for all x ∈ R d and all f ∈ C ∞ c ( R d ) . It iscalled regular if one can write V t = M t + Z t v s ds for a vector field-valued adapted process v • , and an F -valued local martingale M • for whichthere exists a Symm ( d ) -valued process a s ( x, y ) with (cid:10) ∂ αx M • ( x ) , ∂ βy M • ( y ) (cid:11) t = Z t ∂ αx ∂ βy a s ( x, y ) ds for a range of multiindices α, β depending on the regularity assumptions on a s . The pair ( a • , v • ) is called the local characteristics of the semimartingale velocity field V • . A theory of Stratonovich integration can be constructed for making sense of integrals ofthe form Z t V ◦ ds ( x s ) , for some progressively measurable process x • and some regular semimartingale velocityfield V • , as a limit in probability of symmetric Riemann sums. This requires some almostsure regularity properties on the local characteristics ( a • , v • ) of V • , and some almost surebound on R t (cid:12)(cid:12) a s ( x s , x s ) (cid:12)(cid:12) ds and R t (cid:12)(cid:12) v s ( x s ) (cid:12)(cid:12) ds – see e.g. Section 2.3 of [Kun86]. Under theseconditions, the integral Stratonovich equation(2.1) x t = x + Z t V ◦ ds ( x s ) can be seen to have a unique solution started from any point x ∈ R d . Theorem 14 ([LJW84]) . These solutions can be gathered into a semimartingale stochasticflow whose local characteristics are ( a • , v • + c • ) , where the time-dependent vector field c s hascoordinates c is ( x ) := 12 d X j =1 ∂ y j a ijs ( x, y ) (cid:12)(cid:12) y = x in the canonical basis of R d . Conversely, one can associate to any regular stochastic flow ofdiffeomorphisms φ • a semimartingale velocity field V • , with the same local characteristics as φ • , and such that φ • coincides with the stochastic flow generated by the Stratonovich equation x t = x + Z t V ◦ ds ( x s ) − Z t c s ( x s ) ds. The optimal regularity assumptions on the velocity fields and stochastic flows of mapsare given in Theorems 4.4.1 and 4.5.1 of Kunita’s book. We shall use the full strength ofthese two statements in Section 2.4 to identify semimartingale stochastic flows of maps andthe rough flows associated with the lift of the semimartingale velocity fields into a roughdriver.The correspondence between semimartingale stochastic flows and semimartingale veloc-ity fields via an Itô equation of the form x t = x + Z t V ds ( x s ) is exact, with no need to add the drift R t c s ds . We state it here under the above form aswe shall see below that rough flows are naturally associated with Stratonovich differentialequations.The main difficulty in this business is to deal with the local martingale part of the dy-namics, which is where probability theory is really needed. As a consequence, we shall con-centrate our efforts on local martingale velocity fields in the sequel, the remaining changesto deal with regular semimartingale velocity fields being essentially cosmetic. As above,we shall freely identify in the sequel vector fields with first order differential operators. Inorder to keep consistent notations, we shall also denote by Z t α s m ds and Z t α s m ◦ ds the Itô and Stratonovich integrals of an adapted process α s with respect to a local mar-tingale m s . The aim of this section is to give conditions underwhich a local martingale velocity field can be liftedinto a rough driver. Let D be an open connected subset of R d and let M stand for a localmartingale velocity field. We prove in this Section that such a field can be lifted to a roughdriver M = (cid:0) M ts , M ts (cid:1) s t T , with M ts := M t − M s , under regularity and boundednessassumptions on the local characteristic of M . At a heuristic level, if M is differentiablein space, the second level operator M ts associated with M ts = M t − M s is given by theformula M ts = Z ts M us M ◦ du = (cid:18)Z ts M ius ∂ i M k ◦ du (cid:19) ∂ k + (cid:18)Z ts M jus M k ◦ du (cid:19) ∂ jk , with obvious notations for the operators ∂ k and ∂ jk . In the following, we will use thenotation (cid:0) M ts .M ts (cid:1) := (cid:0) DM ts (cid:1) ( M ts ) . As the classical rules of Stratonovich integration give (cid:18)Z ts M jus M k ◦ du (cid:19) ∂ jk = 12 M jts M kts ∂ jk = 12 M ts M ts − (cid:0) M ts .M ts (cid:1) , we see that the operators M ts can be decomposed as M ts = W ts + 12 M ts M ts , (2.2)where W ts = (cid:18)Z ts M ius ∂ i M k ◦ du (cid:19) ∂ k − (cid:0) M ts .M ts (cid:1) = 12 (cid:18)Z ts M ius ∂ i M kdu − M idu ∂ i M kus (cid:19) ∂ k = 12 Z ts (cid:0) M us .M du − M du .M us (cid:1) is a martingale velocity field defined pointwisely by an Itô integral. The proof that theprocess M ts := ( M ts , M ts ) has a modification which is a p -rough driver for every , and < δ . Then, for every < ǫ < δ , thevelocity field M has a modification that is a continuous process with values in C m,ǫ ;we still denote it by M . Furthermore, for each multi-index α , with | α | m , the timevarying random field ∂ αx M is a local martingale velocity field with quadratic variation d (cid:10) ∂ αx M • ( x ) , ∂ αx M • ( y ) (cid:11) t = ∂ αx ∂ αy a t ( x, y ) dt. (3) Let here M and N be two local martingale velocity fields with values in C m,δ . Thentheir joint quadratic variation (cid:10) M • ( x ) , N • ( y ) (cid:11) t has a continuous modification taking values in b C m,ǫ for every ǫ < δ . Furthermore, if m > , this modification satisfies the identity ∂ αx ∂ βy (cid:10) M • ( x ) , N • ( y ) (cid:11) t = (cid:10) ∂ αx M • ( x ) , ∂ βy N • ( y ) (cid:11) t , for all | α | , | β | m . Proof – Cf. Theorem 3.1.1, Theorem 3.1.2 and Theorem 3.1.3 in [Kun90]. (cid:3) These regularity results will be instrumental in the proof of the following intermediateresult. Note that N in equation (2.3) below is seen as a vector field, not a differentialoperator, so (cid:0) M N (cid:1) ( x ) = (cid:0) D x N (cid:1)(cid:0) M ( x ) (cid:1) . Proposition 16. Let M, N : D × [0 , T ] → R be continuous C m,δ -valued local martingale fields,for m ∈ N and δ ∈ (0 , . Assume M is adapted to the filtration generated by N . Then thepointwisely defined Itô integral t Z t (cid:0) M s N ds (cid:1) ( x ) (2.3) has a continuous modification taking values in C m,α process for every α < δ . Moreover, if m > , the derivative is almost surely given by the formula ∂ x i (cid:18)Z t M s N ds (cid:19) = Z t ∂ x i M s N ds + Z t M s ∂ x i N ds . (2.4) Both assertions also hold for the Stratonovich integral. The proof of this result is somewhat lengthy but rests on classical considerations basedon the regularization theorem 15. Proof – Set U t ( x ) := (cid:18)Z t M s N ds (cid:19) ( x ) . This is a continuous local martingale field with joint quadratic variation given by (cid:10) U • ( x ) , U • ( y ) (cid:11) t = Z t M s ( x ) M s ( y ) d (cid:10) N • ( x ) , N • ( y ) (cid:11) s . (2.5)Fix t ∈ [0 , T ] , some compact set K ⊂ D and set g ( x, y ) := h U • ( x ) , U • ( y ) i t . We first consider the case m = 0 . Choose α < δ ′ < δ . Then, for x, x ′ , y, y ′ ∈ K , wehave (cid:12)(cid:12)(cid:12) g ( x, y ) − g ( x ′ , y ) − g ( x, y ′ ) + g ( x ′ , y ′ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:10) U • ( x ) − U • ( x ′ ) , U • ( y ) − U • ( y ′ ) (cid:11) t (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)Z t (cid:0) M s ( x ) − M s ( x ′ ) (cid:1) (cid:0) M s ( y ) − M s ( y ′ ) (cid:1) d (cid:10) N • ( x, ) , N • ( y ) (cid:11) s (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z t (cid:0) M s ( x ) − M s ( x ′ ) (cid:1) M s ( x ′ ) d (cid:10) N • ( x ) , N • ( y ) − N • ( y ′ ) (cid:11) s (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z t M s ( x ′ ) (cid:0) M s ( y ) − M s ( y ′ ) (cid:1) d (cid:10) N • ( x ) − N • ( x ′ ) , N • ( y ) (cid:11) s (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z t M s ( x ′ ) M s ( y ′ ) d (cid:10) N • ( x ) − N • ( x ′ ) , N • ( y ) − N • ( y ′ ) (cid:11) s (cid:12)(cid:12)(cid:12)(cid:12) . For the first integral, we use Kunita’s extended Cauchy-Schwarz inequality, as statedin [Kun90, Theorem 2.2.13], to see that (cid:12)(cid:12)(cid:12)(cid:12)Z t (cid:0) M s ( x ) − M s ( x ′ ) (cid:1) (cid:0) M s ( y ) − M s ( y ′ ) (cid:1) d (cid:10) N • ( x ) , N • ( y ) (cid:11) s (cid:12)(cid:12)(cid:12)(cid:12) (cid:18)Z t (cid:0) M s ( x ) − M s ( x ′ ) (cid:1) d (cid:10) N • ( x ) (cid:11) s (cid:19) (cid:18)Z t ( M s ( y ) − M s ( y ′ )) d (cid:10) N • ( y ) (cid:11) s (cid:19) | x − x ′ | δ | y − y ′ | δ sup s ∈ [0 ,T ] k M s k δ − Höl ; K sup z ∈ K (cid:10) N • ( z ) (cid:11) T . Similarly, for the second integral, (cid:12)(cid:12)(cid:12)(cid:12)Z t (cid:0) M s ( x ) − M s ( x ′ ) (cid:1) M s ( x ′ ) d (cid:10) N • ( x ) , N • ( y ) − N • ( y ′ ) (cid:11) s (cid:12)(cid:12)(cid:12)(cid:12) (cid:18)Z t (cid:0) M s ( x ) − M s ( x ′ ) (cid:1) d (cid:10) N • ( x ) (cid:11) s (cid:19) (cid:18)Z t M s ( x ′ ) d (cid:10) N • ( y ) − N • ( y ′ ) (cid:11) s (cid:19) | x − x ′ | δ | y − y ′ | δ ′ sup s ∈ [0 ,T ] (cid:13)(cid:13) M s (cid:13)(cid:13) ∞ ; K sup s ∈ [0 ,T ] (cid:13)(cid:13) M s (cid:13)(cid:13) δ − Höl ; K sup z ∈ K (cid:10) N • ( z ) (cid:11) T (cid:13)(cid:13)(cid:13)(cid:10) N • , N • (cid:11) T (cid:13)(cid:13)(cid:13) δ ′ ; K . From point 3 in Theorem 15 we know that there is a version of the joint quadraticvariation of N such that (cid:13)(cid:13) h N • , N • i T (cid:13)(cid:13) ∧ δ ′ ; K < ∞ . The other integrals are estimatedsimilarly. This shows that sup x,x ′ ,y,y ′∈ K x = x ′ ,y = y ′ (cid:12)(cid:12) g ( x, y ) − g ( x ′ , y ) − g ( x, y ′ ) + g ( x ′ , y ′ ) (cid:12)(cid:12) | x − x ′ | δ ′ | y − y ′ | δ ′ < ∞ . Clearly k g k ∞ ; K < ∞ , thus we have shown that the joint quadratic variation of U has amodification which is a continuous b C ,δ ′ -process. Point 1 in Theorem 15 shows that U has a modification which is a continuous C ,α -process. Now let m > . From point 3 inTheorem 15 we may deduce that the joint quadratic variation of N has a modificationwhich is a continuous ˜ C m,δ ′ -process with ∂ βx ∂ γy (cid:10) N • ( x ) , N • ( y ) (cid:11) = (cid:10) ∂ βx N • ( x ) , ∂ γy N • ( y ) (cid:11) for every | β | , | γ | m . We may apply Proposition 32 in Appendix iteratively in equa-tion (2.5) to show that (cid:10) U • ( x ) , U • ( y ) (cid:11) t has a modification which is m -times differen-tiable with respect to x and y , and we can calculate the derivatives using the productrule stated in Proposition 32. As above, one can show that the m -th derivative has theclaimed Hölder regularity, and we can conclude with point 2 of Theorem 15 that U hasa modification which is a continuous C m,α -process. The Itô-Stratonovich conversionformula Z t M s N ◦ ds = Z t M s N ds + 12 (cid:10) M • , N • (cid:11) t and point 3 in Theorem 15 show that the same is true for the Stratonovich integral.We now come to equation (2.4). In the following, we use k · k L for the L -norm withrespect to P . For n ∈ N , set τ n = inf n t ∈ [0 , T ] : (cid:13)(cid:13) M t (cid:13)(cid:13) C ,δ + (cid:13)(cid:13) N t (cid:13)(cid:13) C ,δ > n o . The random times τ n define an increasing sequence of stopping times such that P ( τ n Let M, N : D × [0 , T ] → R be continuous local martingale fields adapted tothe same filtration. Assume that the quadratic variation of the processes is given by d h M • ( x ) , M • ( y ) i t = a t ( x, y ) dt resp. d h N • ( x ) , N • ( y ) i t = b t ( x, y ) dt for every x, y ∈ D and every t ∈ [0 , T ] . Moreover, assume that there is a δ ∈ (0 , such that a and b have continuous modifications in the space ˆ C ,δb . Let p > and ρ ∈ (0 , δ ) .(i) Assume that there is a constant κ > and some η ∈ (0 , ∞ ) such that sup u ∈ [0 ,T ] (cid:13)(cid:13)(cid:13)p a u ( x, x ) (cid:13)(cid:13)(cid:13) L q (Ω) κ | x | η (2.7) holds for every x ∈ D where q > max ( δη ( δ − ρ ) + d (cid:18) δη ( δ − ρ ) + 1 δ − ρ (cid:19) , − p ) . (2.8) Then the process M has a modification which satisfies sup s We start with (i). In a first step, we assume that sup t ∈ [0 ,T ] k a t k ∧ δ is an almost surelybounded random variable and that (2.7) holds uniformly for every q > . Under theseassumptions, we aim to show that M has a modification such that sup s Let M be a continuous local martingale velocity field in C ,δ ( D, R d ) with con-tinuous local characteristic a in ˆ C ,δb for some δ ∈ (0 , .(i) Let ρ ∈ (0 , δ ) and p ∈ (2 , . Assume that there is an η ∈ (0 , ∞ ) and a constant κ > such that (2.12) X | α | sup u ∈ [0 ,T ] (cid:13)(cid:13)(cid:13)q ∂ αx ∂ αy a u ( z, z ) (cid:13)(cid:13)(cid:13) L q κ | z | η for every z ∈ D where q satisfies q > max ( δη ( δ − ρ ) + d (cid:18) δη ( δ − ρ ) + 1 δ − ρ (cid:19) , − p ) . Then M = ( M, M ) , M being defined as in (2.2) , has a modification which is a weakgeometric ( p, ρ ) -rough driver. We call M the natural lift of M .(ii) Assume that sup t ∈ [0 ,T ] k a t k ∧ δ is an almost surely bounded random variable and thatthere is an η ∈ (0 , ∞ ) and a constant κ > such that (2.12) holds uniformly for every z ∈ D and every q > .Then for every p ∈ (2 , and ρ ∈ (0 , δ ) , M = ( M, M ) has a modification which isa weak geometric ( p, ρ ) -rough driver, and the random variables sup s The claim for M follows by applying Proposition 17 to M and its derivatives. For W , we use the product rule in Proposition 16 for calculating the derivatives and applyProposition 17 afterwards. The estimates for W together with M yield the claimedestimates for M . We leave the details to the reader. (cid:3) Remark 19. In the special case where M or a have compact support, i.e. when there existsa deterministic compact set K ⊂ D such that M resp. a are supported on K almost surely,assertion (i) of Theorem 18 holds without any moment conditions on a . Indeed, this followsfrom the fact that the stopped processes in Proposition 17 trivially satisfy the growth conditionstated in the Kolmogorov theorem 12, and this was the only point where these assumptionswere needed.2.4. Stochastic and rough flows We keep in this Section the notations of the previous sec-tions, and denote in particular by ( F t ) t T a filtrationto which the semimartingale velocity field M is adapted. Assume that the local character-istic a of M satisfies the boundedness assumptions of point (i) in Theorem 18. Then wecan use Theorem 18 to define the natural lift M of M into a rough driver, and one canmake sense of the rough flow ϕ as pathwise solution to the equation dϕ = M ( ϕ ; dt ) using Theorem 5. It follows from equation (1.3), giving ϕ ts as a limit of compositions of µ ba ’s, that ϕ is a semimartingale stochastic flow of homeomorphisms. One can read itslocal characteristics on the Itô formula that it satisfies. Given x, y ∈ R d and f, g ∈ C b , wehave f (cid:0) ϕ ts ( x ) (cid:1) = f ( x ) + (cid:0) M ts f (cid:1) ( x ) + 12 (cid:26)(cid:16) Z ts M us .M du − M du .M us (cid:17) f (cid:27) ( x ) + 12 (cid:0) M ts f (cid:1) ( x )+ O (cid:16) | t − s | p (cid:17) , with an O ( · ) term depending only on k M k and k f k C , with a similar formula for g (cid:0) ϕ ts ( y ) (cid:1) .We read on this identity that lim h ↓ E " f (cid:0) ϕ t + h,t ( x ) (cid:1) − ϕ ( x ) h (cid:12)(cid:12)(cid:12)(cid:12) F t L = lim h ↓ E (cid:20) (cid:16) M t + h,t f (cid:17) ( x ) (cid:12)(cid:12)(cid:12) F t (cid:21) , and lim h ↓ h E h(cid:8) f (cid:0) ϕ t + h,t ( x ) (cid:1) − f ( x ) (cid:9)(cid:8) g (cid:0) ϕ t + h,t ( y ) (cid:1) − g ( y ) (cid:9)(cid:12)(cid:12)(cid:12) F t i = lim h ↓ h E h(cid:0) M t + h,t f (cid:1) ( x ) (cid:0) M t + h,t g (cid:1) ( y ) (cid:12)(cid:12)(cid:12) F t i = h f, g i t ( x, y ) . So the semimartingale stochastic flow ϕ has the same local characteristics as the semi-martingale stochastic flow generated by the Stratonovich differential equation(2.13) dx t = M ◦ dt ( x t ); they coincide by Theorem 14, such as stated in Theorems 4.4.1 and 4.5.1 in Kunita’s book,as assumption (2.12) on the local characteristic a of M is clearly stronger than the optimalassumptions of Kunita. Theorem 20. Let M be a continuous local martingale velocity field in C ,δ ( R d , R d ) , for some δ ∈ (cid:0) , (cid:3) , with continuous local characteristic a in b C ,δb . Let M be the rough driver associatedwith M by Theorem 18. Under the condition that M or a have compact support or that thegrowth condition (2.12) holds, the rough flow solution to the differential equation dϕ = M ( ϕ ; dt ) coincides with the stochastic flow generated by the Stratonovich differential equation (2.13) .2.5. Strong approximations We give in this Section an example of use of the continuity ofthe Itô map, in the setting of rough drivers and rough flows, byproving a Wong-Zakaï type theorem for semimartingale stochastic flows of maps. That is,we prove that such flows are limits in probability of flows generated by ordinary differentialequations. Granted the continuity of the Itô map, the core of the proof consists in showingthat a rough lift of a continuous piecewise linear time interpolation of a semimartingalevelocity field M converges in probability to M in the topology of rough drivers.As in the last section, let M : [0 , T ] → C ,δ (cid:0) D, R d (cid:1) be a continuous local martingale velocity field with quadratic variation (cid:10) M i • ( x ) , M j • ( y ) (cid:11) t = Z t a ijs ( x, y ) ds and δ ∈ (0 , . Let D = (cid:8) t < t < . . . < t n = T (cid:9) be a partition of the interval [0 , T ] and define the piecewise linear approximation of M with respect to D as M D t := M t i + ( t − t i ) M t i +1 − M t i t i +1 − t i if t ∈ [ t i , t i +1 ] . Note that D 7→ M D commutes with the spatial derivate, i.e. ∂ x i (cid:0) M D (cid:1) = (cid:0) ∂ x i M (cid:1) D =: ∂ x i M D . Define the mesh size of the partition by the formula |D| := max i (cid:12)(cid:12) t i +1 − t i (cid:12)(cid:12) . We define thefirst order differential operator W D ts := 12 (cid:18)Z ts M D ; ius ∂ i M D ; kdu − Z ts M D ; idu ∂ i M D ; kus (cid:19) ∂ k = 12 Z ts ( M D us .M D du − M D du .M D us ) via usual Riemann-Stieltjes integration. Then we set M D ts := W D ts + 12 M D ts M D ts and M D := (cid:0) M D , M D (cid:1) . (2.14)Our aim is to prove that M D converges towards the natural lift M of M when |D| → inprobability (or even in L p (Ω) ) in the topology of rough drivers. Note that W ts = 12 Z ts ( M us .M du − M du .M us ) = 12 Z ts ( M us .M ◦ du − M ◦ du .M us ) , hence it is enough to prove that the Riemann-Stieltjes integrals of the approximated pro-cesses converge towards the Stratonovich integrals (in the right topology), and this is whatwe are going to do. Lemma 21. Let M = (cid:0) M , . . . , M d (cid:1) : [0 , T ] → R d be a continuous local martingale andassume that (cid:13)(cid:13) h M i T (cid:13)(cid:13) L q K < ∞ for some q > and K > . Let M and M D be the associated rough paths lifts to M and M D , i.e. M = ( M, M ) and M D = (cid:0) M D , M D (cid:1) where M = Z ts M us M ◦ du and M D = Z ts M D us M D du are iterated Stratonovich, resp. Riemann-Stieltjes, integrals. Set ε := (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup Let d denote the Carnot-Caratheodoy metric on the step-two free nilpotent Liegroup G d over R d [cf. [FV10, Chapter 7]). By interpolation and [FV10, Proposition8.15], for fixed s < t and < p ′ < p < , d ( M ts , M D ts ) (cid:18) d − Höl ( M , M D ) − p ′ p d p ′ − var ;[ s,t ] ( M , M D ) p ′ p (cid:19) . d ∞ ( M , M D ) − p ′ p + d ∞ ( M , M D ) − p ′ p ( k M k − p ′ p ∞ + k M D k − p ′ p ∞ ) ! × (cid:16) k M k p ′ /pp ′ − var ;[ s,t ] + k M D k p ′ /pp ′ − var ;[ s,t ] (cid:17) . Taking the q -th moment, using Hölder’s inequality and Cauchy-Schwarz, we obtain E h d (cid:0) M ts , M D ts (cid:1) q i . E h d ∞ (cid:0) M , M D (cid:1) q i + r E h d ∞ (cid:0) M , M D (cid:1) q i (cid:16) E (cid:2) k M k q ∞ (cid:3) + E (cid:2) k M D k q ∞ (cid:3)(cid:17)! − p ′ p × (cid:16) E h k M k qp ′ − var ;[ s,t ] i + E h k M D k qp ′ − var ;[ s,t ] i(cid:17) p ′ p . By [FV10, Theorem 14.8 and Theorem 14.15], E (cid:2) k M k q ∞ (cid:3) . E h h M i T | q i and E (cid:2) k M D k q ∞ (cid:3) E h k M D k qp − var i . E h h M i T | q i . Using the same theorems, we also have E h k M k qp ′ − var ;[ s,t ] i . E h h M i q ts i and E h k M D k qp ′ − var ;[ s,t ] i . E h h M i q ts i . The estimate [FV10, Equation (14.6) on p. 400] gives E h d ∞ (cid:0) M , M D (cid:1) q i . E h sup Let M, N : D × [0 , T ] → R be continuous local martingale fields adapted tothe same filtration. Assume that the quadratic variation of the processes is given by d h M • ( x ) , M • ( y ) i t = a t ( x, y ) resp. d h N • ( x ) , N • ( y ) i t = b t ( x, y ) for every x, y ∈ D and every t ∈ [0 , T ] . Moreover, assume that there is a δ ∈ (0 , such that a and b have continuous modifications in the space ˆ C ,δb . Let p > and ρ ∈ (0 , δ ) .(i) Assume that there is an η ∈ (0 , ∞ ) and a constant κ > such that sup u ∈ [0 ,T ] (cid:13)(cid:13)(cid:13)p a u ( x, x ) (cid:13)(cid:13)(cid:13) L q κ | x | η for every x ∈ D where q > max ( δη ( δ − ρ ) + d (cid:18) δη ( δ − ρ ) + 1 δ − ρ (cid:19) , − p ) . (2.15) Let M be the modification of the process given in Proposition 17.Then sup s We will only give a proof in the case of a and b having compact support. Themore general case works analogous using the stated growth conditions, as seen in theproof of Proposition 17. We start with (i). Assume first that sup u ∈ [0 ,T ] k a u k ∧ δ is analmost surely bounded random variable. Fix x, y ∈ D such that < | x − y | andlet s < t ∈ [0 , T ] . Define the martingale ˆ M := M ( x ) − M ( y ) and let ˆ M denote itscanonical rough path lift (given by ˆ M and its iterated Stratonovich integrals) . Fromthe Burkholder Davis Gundy inequality for enhanced martingales (cf. [FV10, Theorem14.8]), for every q > , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup Let M be a continuous local martingale velocity field in C ,δ ( D, R d ) , for some δ ∈ (0 , , with continuous local characteristic a in ˆ C ,δb . (i) Let ρ ∈ (0 , δ ) and p ∈ (2 , . Assume that there is an η ∈ (0 , ∞ ) and a constant κ > such that (2.21) X | α | sup u ∈ [0 ,T ] (cid:13)(cid:13)(cid:13)q ∂ αx ∂ αy a u ( z, z ) (cid:13)(cid:13)(cid:13) L q κ | z | η for every z ∈ D where q satisfies q > max ( δη ( δ − ρ ) + d (cid:18) δη ( δ − ρ ) + 1 δ − ρ (cid:19) , − p ) . Let M = ( M, M ) be the weak geometric ( p, ρ ) -rough driver given in Theorem 18.Then M D → M (2.22) in probability for |D| → , with M D defined as in (2.14) .(ii) Assume that sup u ∈ [0 ,T ] k a u k ∧ δ is almost surely bounded, and that the growth condition (2.21) holds for some η ∈ (0 , ∞ ) and a constant κ > uniformly for all q > .Then for every p ∈ (2 , and ρ ∈ (0 , δ ) , M = ( M, M ) has a modification whichis a weak geometric ( p, ρ ) -rough driver, and the convergence in (2.22) holds in L q forevery q > . Proof – This follows by applying Proposition 22 to M , W ,and M D and W D and its deriva-tives. We use the product rule in Proposition 16 for calculating the derivatives of W and Proposition 32 for the derivatives of W D . The details are left to the reader. (cid:3) Remark 24. As seen in the proof of Proposition 22, assertion (i) in Theorem 23 holds withoutany moment conditions on a in the case where M or a have compact support. It follows then directly from this statement and the continuity of the Itô solution map,Theorem 5, that the solution flow to the equation dϕ = M ( ϕ ; dt ) satisfies a Wong-Zakaï theorem. Using that the flow coincides with the one generated bythe corresponding Stratonovich SDE, we obtain the following corollary. Corollary 25. Let M be a continuous local martingale velocity field in C ,δ ( R d , R d ) , for some δ ∈ (cid:0) , (cid:3) , with local characteristic a in b C ,δb . Assume that M or a have compact support orthat the growth assumption (2.12) holds. Let ϕ be the flow generated by the Stratonovichsolution to dϕ = M ( ϕ ; ◦ dt ) and ϕ D be the pathwise solution to dϕ D = M D (cid:0) ϕ D ; dt (cid:1) . Then ϕ D → ϕ in the space of C ρ homeomorphisms uniformly in probability when |D| → forall ρ ∈ (0 , δ ) . Application: Large deviations We provide in this Section another illustration of use of the continuity of the Itô map byproving a large deviation theorem for Brownian flows. Relatively few works were dedicatedto these topices, and we mention [BDM10] and [DD12]. In [BDM10], Dupuis and his co-authors use Dupuis’ weak convergence approach to large deviation principles to prove sucha result for Brownian flows of maps, building on a general large deviation criterion provedearlier in [BDM08]. Dereich and Dimitroff’s approach in [DD12] is more in the line ofthe present work. They consider Brownian flows of maps as solutions to rough differentialequations driven by a Banach space valued Brownian rough path, whose construction in avector field setting was made possible by the previous work [Der10] of Dereich. The supportand large deviation theorems for Brownian flows are then inherited from the correspondingresults proved in [Der10] for the above mentioned vector field-valued Brownian roughLet ( E, H , γ ) be a Gaussian Banach space with norm k · k , i.e. ( E, k · k ) is a separableBanach space, γ is a Gaussian measure defined on the Borel σ algebra and H the Cameron-Martin Hilbert space (cf. [Bog98] or [Led96, Chapter 4] for the precise definitions andfurther properties). Recall that H is continuously embedded in E , and for every h ∈ H , k h k σ γ k h k H = σ γ p h h, h i H where σ γ := Z E k x k γ ( dx ) . A process X : [0 , T ] → E , defined on some probability space, is called a E -valued Wienerprocess if it has almost surely continuous sample paths starting from , has independentincrements, and for every ξ ∈ E ∗ , the distribution of h X t − X s , ξ i is a centered Gaussianrandom variable with variance | t − s || ξ | H (cf. [LLQ02] and [Der10] for more propertiesabout E -valued Wiener processes). The law of X on the space C (cid:0) [0 , T ] , E (cid:1) is again Gauss-ian, and one can see that the corresponding Cameron-Martin space H is given by H = (cid:26)Z • f s ds : f ∈ L (cid:0) [0 , T ] , H (cid:1)(cid:27) where the integral is a Bochner-integral. Moreover, if h it = R t ˙ h is ds , i = 1 , , the scalarproduct is given by (cid:10) h , h (cid:11) H = Z T (cid:10) ˙ h s , ˙ h s (cid:11) H ds. Let p ∈ [1 , ∞ ) . In the following, we will use the notion of p -variation of a path h : [0 , T ] → E which is defined as k h k p − var ;[ s,t ] := sup ( t i ) ⊂ [ s,t ] X t i (cid:13)(cid:13) h t i +1 − h t i (cid:13)(cid:13) p ! p where the supremum is taken over all finite partitions ( t i ) of the interval [ s, t ] . If k h k − var ;[0 ,T ] < ∞ , we say that h has finite variation. Lemma 26. For every h ∈ H we have sup s Clearly, (3.1) follows from (3.2), hence we only prove the second estimate. Let h ∈ H with h ( t ) = R t ˙ h s ds and let ( t i ) be a partition of some interval [ s, t ] ⊆ [0 , T ] .Then X i k h t i +1 − h t i k X i Z t i +1 t i k ˙ h u k du = Z ts k ˙ h u k du | t − s | (cid:18)Z T k ˙ h u k du (cid:19) σ γ | t − s | p h h, h i H . Taking the supremum over all partitions shows the claim. (cid:3) Let D be a an open, relatively compact, connected subset in R d , m ∈ N and δ ∈ (0 , . In the following, we would like to take the space C m,δb ( D, R d ) as E and consider aGaussian measure on this space. However, C m,δb ( D, R d ) is not separable (which is usuallythe case for Hölder-type spaces). Instead, we define the space C m, ,δb ( D, R d ) as the closureof smooth paths from D to R d with respect to the norm k · k m + δ . As for Hölder spaces,using boundedness of D , one can show that these spaces are separable. From now on, let E = C m, ,δb ( D, R d ) and assume that there is a Gaussian Banach space ( E, H , γ ) .If v is a C m,δb ( D, R d ) valued path with finite variation and if m > , we define the pair S ( v ) ts = ( v ts , v ts ) by setting v ts ( x ) = v t ( x ) − v s ( x ) and v ts = w ts + 12 v ts v ts where w ts is the first order differential operator w ts = 12 (cid:18)Z ts v us .v du − v du .v us (cid:19) and the integral is a Riemann-Stieltjes integral. Note that if X is a Wiener process in C m, ,δb , S ( h ) is always defined for every Cameron-Martin path h since these paths arecontinuous and have bounded variation by Lemma 26. Moreover, the following holds: Lemma 27. Let ( E, H , γ ) be a Gaussian Banach space with E = C , ,δb ( D, R d ) and δ ∈ (0 , .Then, for every h ∈ H , S ( h ) is a geometric (2 , δ ) -rough driver, and there is a constant C suchthat sup s Let h ∈ H and S ( h ) = ( h, h ) be defined as above. The claim for h followsdirectly from Lemma 26, and the algebraic condition for ( h, h ) follows from well-known identities for Riemann-Stieltjes integrals. Let i, k ∈ { , . . . , d } , x ∈ D and s < t . Then, by Riemann-Stieltjes estimates and Lemma 26, (cid:12)(cid:12)(cid:12)(cid:12)Z ts h ius ( x ) ∂ i h kdu ( x ) (cid:12)(cid:12)(cid:12)(cid:12) sup u ∈ [ s,t ] (cid:12)(cid:12) h ius ( x ) (cid:12)(cid:12) sup ( t j ) ⊂ [ s,t ] X j (cid:12)(cid:12) ∂ i h kt j +1 ( x ) − ∂ i h kt j ( x ) (cid:12)(cid:12) sup u ∈ [ s,t ] k h us k δ sup ( t j ) ⊂ [ s,t ] X j (cid:13)(cid:13) h t j +1 − h t j (cid:13)(cid:13) δ σ γ | t − s |h h, h i . One can perform the same estimate for the second term in w ts . By the triangleinequality, this shows that sup s Let D be a relatively compact domain in R d and X be a Wiener process in C , ,δb for some δ ∈ (0 , . Let X = ( X, X ) denote its natural lift to a ( p, ρ ) -rough driver for some ρ ∈ (0 , δ ) and p ∈ (2 , . Let η > be fixed. Then the following holds: lim n →∞ lim sup ε → ε log P (cid:16) d p,ρ ( δ ε X , δ ε X n ) > η (cid:17) = −∞ . Proof – Let ε > and n ∈ N be fixed. Since X is Gaussian, the quadratic variationprocess a is deterministic, and all estimates for X , X n , X − X n and its iteratedintegrals in the proof of Proposition 22 hold for q = 2 . Moreover, | X | L q . √ q | X | L .The iterated Stratonovich and Riemann-Stieltjes integrals are both elements in thesecond inhomogeneous Wiener chaos, therefore (cid:13)(cid:13)(cid:13)(cid:13)Z ◦ dX ⊗ ◦ dX (cid:13)(cid:13)(cid:13)(cid:13) L q . q (cid:13)(cid:13)(cid:13)(cid:13)Z ◦ dX ⊗ ◦ dX (cid:13)(cid:13)(cid:13)(cid:13) L and (cid:13)(cid:13)(cid:13)(cid:13)Z dX n ⊗ dX n (cid:13)(cid:13)(cid:13)(cid:13) L q . q (cid:13)(cid:13)(cid:13)(cid:13)Z dX n ⊗ dX n (cid:13)(cid:13)(cid:13)(cid:13) L , for all q > , see e.g. [FV10, Theorem D.8], and similar estimates hold for the otherquantities. Therefore, we may apply Theorem 13 with κ equal to a constant times √ q which shows that (cid:13)(cid:13) d p,ρ ( X , X n ) (cid:13)(cid:13) L q = α n √ q holds for all q > and some constant α n . Repeating this argument for every n ∈ N ,we obtain a sequence ( α n ) converging to for n → ∞ . Thus P (cid:0) d p,ρ ( δ ε X , δ ε X n ) > η (cid:1) = P (cid:16) d p,ρ ( X , X n ) > ηε (cid:17) (cid:18) εη (cid:19) q q q α qn exp (cid:20) q log (cid:18) εα n √ qη (cid:19)(cid:21) . Choosing q = ε − we obtain the inequality ε log P (cid:0) d p,ρ ( δ ε X , δ ε X n ) > η (cid:1) log( α n /η ) from which the claim follows. (cid:3) If H is the Cameron-Martin space of a C , ,δb -valued Wiener process and v is a path withvalues in C , ,δb , set I ( v ) := ( h v, v i H , if v ∈ H + ∞ otherwise. Lemma 29. Let H be the Cameron-Martin space for some C , ,δb -valued Wiener process.Choose Λ > . Then lim |D|→ sup { h ∈H : I ( h ) Λ } d p,δ (cid:16) S ( h D ) , S ( h ) (cid:17) = 0 for every p > . Proof – It is easy to check (cf. [FV10, Proposition 5.20] and Lemma 27) that sup s The proof is standard, using the large deviation principle for Gaussian measure[DS89, Section 3.4], the extended contraction principle [DZ98, Theorem 4.2.23] andthe results in the Lemmas 28 and 29 (cf. e.g. [FV10, Theorem 13.42]). (cid:3) As an immediate corollary, we obtain Freidlin-Ventzel large devations for a class ofstochastic flows. Theorem 31. Let X be a Wiener process in C , ,δb ( D, R d ) , for some δ ∈ (cid:0) , (cid:3) , and let ϕ ε bethe flow generated by the Stratonovich solution to dϕ ε = εX ( ϕ ε ; ◦ dt ) . Let ν ε denote the law of ϕ ε in the space of C ρ homeomorphisms, ρ ∈ (cid:0) , δ (cid:1) . Then the family { ν ε : ε > } of probability measures satisfies a large deviation principle with speed ε − andgood rate function L ( ψ ) = inf n J ( v ) : dψ = v ( ψ ; dt ) o . Proof – The Stratonovich solution equals the solution generated by the ( p, ρ ) -rough driver X . Using Theorem 30 and the pathwise continuity X ϕ , we can use the usualcontraction principle in large deviation theory [DZ98, Theorem 4.2.1] to conclude. (cid:3) Appendix We provide in this Appendix an elementary regularity result for integrals depending ona parameter.Let δ ε be a standard Dirac sequence. If I is a closed interval and f : I → R is a continuousfunction, let ¯ f : R → R denote the unique continuous extension which coincides with f on I and which is constant outside this interval. Set f ε := δ ε ∗ ¯ f . If D is some subset of R d and if f : D × I → R is a continuous function in time for every x ∈ D , set f ε ( x, t ) := (cid:0) δ ε ∗ ¯ f ( x, · ) (cid:1) ( t ) . Proposition 32. Let D ⊂ R d be an open set and let f : D × [0 , T ] → R and g : D × [0 , T ] → R be continuous. Assume that f and g are continuously differentiable on D and that f ( x, 0) = ∂ x i f ( x, 0) = 0 for every x ∈ D and every i = 1 , . . . , d . Moreover, assume that there are p, q ∈ [1 , ∞ ) with p + q > such that sup ( t i ) X t i (cid:13)(cid:13) f ( · , t i +1 ) − f ( · , t i ) (cid:13)(cid:13) p C and sup ( t i ) X t i (cid:13)(cid:13) g ( · , t i +1 ) − g ( · , t i ) (cid:13)(cid:13) q C are finite, where the suprema are taken over all finite partitions of the interval [0 , T ] . Then theYoung integral (cf. e.g. [FV10, Chapter 6] for the precise definition) R T f ( x, t ) g ( x, dt ) exists,is continuously differentiable for all x ∈ D and the derivative is given by ∂ x i (cid:18)Z T f ( x, t ) g ( x, dt ) (cid:19) = Z T ∂ x i f ( x, t ) g ( x, dt ) + Z T f ( x, t ) ∂ x i g ( x, dt ) for all i = 1 , . . . , d . Proof – One can suppose without loss of generality that i = 1 . Fix x ∈ D . ∂ x (cid:18)Z T f ( x, t ) g ε ( x, dt ) (cid:19) = ∂ x (cid:18)Z T f ( x, t ) ∂ t ( g ε ( x, t )) dt (cid:19) = Z T ∂ x f ( x, t ) ∂ t ( g ε ( x, t )) dt + Z T f ( x, t ) ∂ t (( ∂ x g ( x, · )) ε ( t )) dt = Z T ∂ x f ( x, t ) g ε ( x, dt ) + Z T f ( x, t )( ∂ x g ( x, · )) ε ( dt ) . Let q ′ > q such that p + q ′ > . Let U be some neighbourhood of x and let y ∈ U .From Young estimates and interpolation, we obtain (cid:12)(cid:12)(cid:12)(cid:12)Z T ∂ y f ( y, t ) g ε ( y, dt ) − Z T ∂ y f ( y, t ) g ( y, dt ) (cid:12)(cid:12)(cid:12)(cid:12) C sup ( t i ) ⊂ [0 ,T ] X t i (cid:12)(cid:12) ∂ x f ( y, t t i +1 ) − ∂ x f ( y, t i ) (cid:12)(cid:12) p ! p × sup ( t i ) ⊂ [0 ,T ] X t i (cid:12)(cid:12) g ε ( y, t i +1 ) − g ( y, t i +1 ) − g ε ( y, t i ) + g ( y, t i ) (cid:12)(cid:12) q ′ ! q ′ C sup ( t i ) ⊂ [0 ,T ] X t i (cid:13)(cid:13) f ( · , t i +1 ) − f ( · , t i ) (cid:13)(cid:13) p C ! p × sup ( t i ) ⊂ [0 ,T ] X t i (cid:13)(cid:13) g ε ( · , t i +1 ) − g ε ( · , t i ) (cid:13)(cid:13) q C ! q + sup ( t i ) ⊂ [0 ,T ] X t i (cid:13)(cid:13) g ( · , t i +1 ) − g ( · , t i ) (cid:13)(cid:13) q C ! q qq ′ × − qq ′ sup t T (cid:13)(cid:13) g ε ( · , t ) − g ( · , t ) (cid:13)(cid:13) − qq ′ C . It is easy to check that sup ( t i ) ⊂ [0 ,T ] X t i (cid:13)(cid:13) g ε ( · , t i +1 ) − g ε ( · , t i ) (cid:13)(cid:13) q C ! q sup ( t i ) ⊂ [0 ,T ] X t i (cid:13)(cid:13) g ( · , t i +1 ) − g ( · , t i ) (cid:13)(cid:13) q C ! q . Therefore, we obtain a bound of the form (cid:12)(cid:12)(cid:12)(cid:12)Z T ∂ y f ( y, t ) g ε ( y, dt ) − Z T ∂ y f ( y, t ) g ( y, dt ) (cid:12)(cid:12)(cid:12)(cid:12) C sup t T (cid:13)(cid:13) g ε ( · , t ) − g ( · , t ) (cid:13)(cid:13) − β ′ /β C where C is independent of y and ε . Thus, Z T ∂ x f ( y, t ) g ε ( y, dt ) → Z T ∂ x f ( y, t ) g ( y, dt ) uniformly in a neighbourhood around x when ε → . Similarly, Z T f ( y, t )( ∂ x g ( y, · )) ε ( dt ) → Z T f ( y, t ) ∂ y g ( y, dt ) uniformly in a neighbourhood around x when ε → . This shows differentiability in x of the integral Z T f ( x, t ) g ( x, dt ) and the claimed identity. (cid:3) References [Bai15] Ismael Bailleul. Flows driven by rough paths. Rev. Mat. 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