H\"ormander's theorem for semilinear SPDEs

We consider a broad class of semilinear SPDEs with multiplicative noise driven by a finite-dimensional Wiener process. We show that, provided that an infinite-dimensional analogue of H\"ormander's bracket condition holds, the Malliavin matrix of the solution is an operator with dense range. In particular, we show that the laws of finite-dimensional projections of such solutions admit smooth densities with respect to Lebesgue measure. The main idea is to develop a robust pathwise solution theory for such SPDEs using rough paths theory, which then allows us to use a pathwise version of Norris's lemma to work directly on the Malliavin matrix, instead of the"reduced Malliavin matrix"which is not available in this context. On our way of proving this result, we develop some new tools for the theory of rough paths like a rough Fubini theorem and a deterministic mild It\^o formula for rough PDEs.


Introduction
The goal of this paper is to generalise the series of articles [HM , BM , HM ] where the authors developed Malliavin calculus for semilinear stochastic partial differential equations (SPDEs) with additive degenerate noise and showed non-degeneracy of the Malliavin matrix under Hörmander's bracket condition. The main novelty of the present article is that we are able to extend these results to equations driven by I multiplicative noise. In particular, we conclude that finite-dimensional projections of the solutions admit densities and, provided that suitable a priori bounds are satisfied, that the corresponding Markov semigroup satisfies the asymptotic strong Feller property introduced in [HM ]. This can be understood as a genuinely infinite-dimensional "smoothing property" for the Markov semigroup that holds at infinite time.
Equations considered in this article can formally be written in Stratonovich form as where L is a negative definite selfadjoint operator on a separable Hilbert space H, N, F i are smooth non-linearities, and B t " pB 1 t , B 2 t , . . . , B d t q is a standard d-dimensional Brownian motion. We potentially allow N to lose derivatives and can consider for instance the D Navier-Stokes equations. Reaction-diffusion equations and the Cahn-Hilliard equation also fall in the category of equations that we consider. Setting F 0 puq " Lu`N puq, we will make regularity assumptions guaranteeing that all iterated Lie brackets of the F i can be given a canonical meaning, so that we can formulate an infinite-dimensional version of Hörmander's condition. Recall that Lie brackets are formally given by rF i , F j spuq " DF j puqF i puq´DF i puqF j puq.
While the Malliavin matrix is not invertible on the whole space H, we will show that it is invertible on every finite-dimensional subspace. (See [DPZ ] for the exceptional case when Malliavin matrix of the linear equation is invertible on the whole space, see also [EH , DPEZ , FM , Cer ] for situations where the Malliavin matrix is invertible on the image of the Jacobian.) Note that the situation considered in this article is orthogonal to the one considered in [HM , CFG ]. There, the authors considered a situation in which the noise already acts in a "full" way in every direction of the state space, so that no Lie brackets need to be considered. Instead, the problem addressed there is that solutions can be very singular, so that sophisticated solution theories need to be considered, which do not interplay nicely with Malliavin calculus.
We now state the main result of the present article: Hörmander's theorem. Let T ą 0 let 0 ď δ ă 2{3 and N : H Ñ H´δ, and F i : H Ñ H be C 8 vector fields of polynomial type satisfying Hörmander's condition, Assumption A. . Assume that u is a global mild solution of the equation ( . ) such that both }u} L 8 pr0,T s,Hq and }J} L 8 pr0,T s,LpH,Hqq have moments of all orders. (here J is Jacobian of the solution). Then for every finite rank orthogonal projection Π : H Ñ H, a P p0, 1q and every p ě 1 there exist a constant C p such that the Malliavin matrix M T satisfies the following bound for every initial condition u 0 : Pˆinf }Πϕ}ąa}ϕ} xM T ϕ, ϕy }ϕ} 2 ď ε˙ď C p ε p .
Moreover the law of Πu T has a smooth density with respect to Lebesgue measure on ΠpHq.
The classical approach to proving a statement of this type was initiated by Malliavin in [Mal ] and further developed and refined by a number of authors in the eighties [Bis , KS , KS , KS , Nor ]. See also [Mal , Nua , Hai ] for surveys of a more expository nature. The argument goes by contradiction: assume that xM T ϕ, ϕy is small (in a suitable probabilistic sense) and use this as the starting point for a chain of implications that eventually lead to an impossibility, resulting in the conclusion that I to use an alternative to Norris's lemma -a certain non-degeneracy bound on Wiener polynomials -but this approach seems to be of little use in the case of multiplicative noise. The idea implemented in the present article is to use the theory of rough paths in order to give meaning to ( . ) directly and to be able to exploit the 'deterministic' version of Norris's lemma for rough paths from [HP ].
The theory of rough paths provides a pathwise approach to a stochastic integration and was originally developed by Lyons [Lyo , LQ ] building on the works of Young and Chen [You , Che ]. The idea is that, in order to solve (finite-dimensional) equations of the type dY t " F pY t qdX t with X P C γ for γ ă 1{2, one augments X with a function X t,s for t ě s that postulates the values of the integrals ş t s δX r,s dX r (we write δX t,s " X t´Xs ) and that satisfies the bound |X t,s | À |t´s| 2γ consistent with the regularity of X, as well as the algebraic identity X t,s´Xu,s´Xt,u " δX t,u bδX u,s . The pair pX, Xq is then called a rough path. Once X is given, integrals of the form ş Y t dX t can be defined in a consistent way for a class of integrands Y that locally "look like X at small scales", see [Gub ] where this notion was introduced. Formally, this can be expressed as δY t,s " Y 1 s δX t,s`R Y t,s , where Y 1 P C γ is the 'Gubinelli derivative' of Y and the remainder R Y satisfies |R Y t,s | À |t´s| 2γ . One then sets where P denotes a partition of r0, T s by intervals and |P| the length of its largest element. It turns out that there exists a canonical lift X Þ Ñ pX, Xq for a wide range of stochastic processes, including Brownian motion (with X defined by Stratonovich integration), fractional Brownian motion and other Gaussian processes.
In [GT ] Gubinelli and Tindel generalised theory of rough paths to solve not only SDEs but also SPDEs: evolution equations driven by the infinite dimensional Gaussian process. For that, they introduce operator-valued rough paths and use a slightly different kind of local (in time) expansion of the controlled processes, taking into account the solution to the linearised equation. This means that we no longer compare Y t to Y s at small scales, but instead to e Lpt´sq Y s . More formally, we replace ( . ) by an expansion of the type Y t´e Lpt´sq Y s " e Lpt´sq Y 1 s δX t,s`R Y t,s . ( . ) Since in our case the driving noise is finite-dimensional, we use similar ideas to [GT ], but then stick closely to the classical theory of finite-dimensional rough paths as in [FH ]. The main difference and complication arises when one wants to show that if Y satisfies an expansion like ( . ) then so does F pY q for any smooth enough function F . This requires an estimate on F pY t q´e Lpt´sq F pY s q while only having a good bound on F pY t q´F pe Lpt´sq Y s q, thus requiring commutator bounds of the type }e Lpt´sq F pY s q´F pe Lpt´sq Y s q} À |t´s| 2γ , which is possible for instance if Y s itself has better space regularity. We therefore need to obtain bounds on the space regularity of the path Y s that are better than the space regularity in which we measure the rough path norms.
One of the main technical difficulties we encounter is to prove that ( . ) holds. An obstacle is that we cannot simply differentiate xJ T,s Gpu s q, ϕy because rough path theory only allows us to use a mild formulation of the solution to ( . ). This however turns out to be sufficient once we obtain a rough Fubini Theorem and a mild version of Itô's formula for Gpu s q. Once we obtain ( . ), we follow closely the approach from [HP ], making use of the rough Norris lemma. We try in most cases to work with general rough paths, not just the one lifted from Brownian motion, so that part of our results carry over immediately to SPDEs driven by fractional Brownian motion for example. We do however show that in the Brownian case the solutions constructed here coincide with those obtained from Itô calculus, which connects our result with existing objects and allows us to exploit information known for the solutions to Itô SPDEs like Malliavin differentiability, a priori bounds and global existence. Such information might be much harder to obtain for more general Gaussian rough paths. We want to emphasise again that once we translate our problem to the language of rough paths, most of the arguments are deterministic. We will only use probabilistic tools (and very basic ones at that) in the proof of Hörmander's theorem itself and in order to obtain global well-posedness of solutions.
Outline of the article: In Section , we introduce a reduced incrementδ and reduced Hölder spaces as well as a version of the sewing lemma from [GT ] for this reduced increment. Section gives a self-contained introduction to the spaces of controlled rough paths with the semigroup and how composition with regular functions preserves these spaces. We also describe an integration in these spaces with respect to rough path which follows directly from the sewing lemma. Section is devoted to the solution theory, continuity of the solution map and the properties of the solution. In particular, we show in Section . that the solutions obtained by viewing ( . ) as an RPDE and driving it by the Stratonovich lift of Brownian motion coincide almost surely with the solutions constructed using classical stochastic calculus as in [DPZ ] for example. Section is about the proof of rough Fubini theorem. In Section we show equivalence of mild solutions and weak solutions. We later use this in order to show a mild Itô formula. Section talks about the backwards equations. There we provide an equation for the adjoint of the Jacobian and also prove the differentiation statement ( . ). Finally in Section we recall the "rough Norris lemma" and combine it with the previous results to prove in Theorem . the Hörmander-type theorem announced in the introduction. We also show in Theorem . how this immediately yields smooth densities for finitedimensional marginals of the solution.

Preliminaries . Semigroup theory
Throughout this paper we consider a separable Hilbert space H with inner product x¨,¨y and a negative definite selfadjoint operator L such that there exists some constant c ă 0 such that xu, Luy ď cxu, uy. We write pS t q tě0 as well as e Lt for the semigroup generated by L. For α ě 0, the interpolation space H α " Dompp´Lq α q is a Hilbert space when endowed with the norm }¨} Hα " }p´Lq α¨} H . Since p´Lq´α is bounded on H, }¨} Hα is equivalent to the graph norm of p´Lq α . Similarly, H´α is defined as the completion of H with respect to the norm }¨} H´α " }p´Lq´α¨} H .

P
For any α, β P R, denote the space of bounded operators from H α to H β by L α,β :" LpH α ; H β q and write L α :" L α,α . We define a reduced semigroup operator S t " S t´i d.
Defining H 8 " Ş α H α and H´8 " Ť α H α the operators S t map H´8 to H 8 for every t ą 0 and H 8 Ă H´8 densely. We will use extensively the fact that, for every α ě β and every γ P r0, 1s, one has }S t u} Hα À t β´α }u} H β , uniformly over t P p0, 1s and u P H β . For an introduction to analytic semigroup theory, see for example [Paz , Hai ].

. Increment spaces
We now define spaces of time increments of functions taking values in some Banach space. We follow closely the definitions in [GT , DGT , DT ]. Fix T ą 0 and, for n P N, define the n-simplex ∆ n " tpt 1 , . . . , t n q : T ě t 1 ě t 2 ě . . . ě t n ě 0u. We will often omit the fact that spaces depend on T since its precise value is not relevant.
Definition . . Given a Banach space V , n P N and T ą 0 define C n pV q :" Cp∆ n , V q the space of continuous functions from ∆ n to V and δ : C n´1 pV q Ñ C n pV q by wheret i indicates that the corresponding argument is omitted.
We are mostly going to use the two special cases δf t,s " f t´fs , δg t,u,s " g t,s´gt,u´gu,s .
One can check that δδ " 0 as an operator C n´1 pV q Ñ C n`1 pV q and that for each f P C n pV q such that δf " 0 there exists g P C n´1 pV q such that f " δg.

Definition . .
For V either the space L α,β or H α for α, β P R, the reduced increment operatorδ : C n´1 pV q Ñ C n pV q is given byδf " δf´Sf , where pSf q t1¨¨¨tn " S t1´t2 f t2¨¨¨tn withS t " S t´i d.
Again, the two most common cases will bê δf t,s " f t´St´s f s ,δg t,u,s " g t,s´gt,u´St´u g u,s .
Whenever we talk aboutδ on C n we will assume from now on that the underlying space V is one of the spaces on which the action of the semigroup S makes sense. Similarly to δ, one verifies thatδδ " 0 and thatδf " 0 implies that f "δg (see [GT ]).

. Hölder type spaces
Definition . . Let V be a Banach space and denote by }¨} V the corresponding norm. Then, for γ, µ ą 0, n ě 2 and f P C n pV q, we set We then define the spaces and notations C γ n " tf P C n : |f | γ,V ă 8u ,Ĉ γ,µ n " tf P C γ n :δf P C µ n`1 u , C γ " tf P C 1 : δf P C γ 2 u ,Ĉ γ " tf P C 1 :δf P C γ 2 u .

P
Since ( . ) doesn't make any sense for n " 1, we make an abuse of notation by writing |f | γ,V for |δf | γ,V for f P C γ . Later on, it will be clear from context whether we use |¨| γ,V as in ( . ) or as the seminorm on C γ . Similarly, we define a seminorm onĈ γ by }f } γ,V " |δf | γ,V and we endow C 1 with the supremum norm }f } 8,V " sup 0ďsďT }f s } V . Finally we equip C γ andĈ γ with norms }f } In the case V " H α , we will write }f } γ,Hα " }f } γ,α , }f } 8,Hα`2γ " }f } 8,α`2γ , etc. An important feature of elements Ξ PĈ γ,µ 2 V is that they can be "integrated" in the sense that Ξ t,s 'almost' looks likeδF t,s for some function F PĈ γ V . More precisely, one has the following version of the sewing lemma.
Theorem . (Sewing Lemma). Let α P R and let 0 ă γ ď 1 ă µ. Then there exist a unique continuous linear map I : then for every β P r0, µq the following inequality holds: Finally, one has the identity where |P| denotes the length of the largest element of a partition P of rs, ts into non-overlapping closed intervals. The same is true if we replace H α by H n α .
Proof. The proof is almost identical to that of [FH , Lemma . ], so we only focus on the details that differ. We first show that the limit ( . ) exists over the dyadic partition: let P 0 " trs, tsu and recursively set P n`1 " ď ru,vsPPn tru, ms, rm, vsu , where m " pu`vq{2, so that P n contains 2 n intervals of length 2´n|t´s|. We then define an approximation to IΞ by: with m the midpoint between u and v as before. We focus on ( . ) under assumption ( . ) since showing ( . ) is even closer to [FH , Lemma . ]. We assumê δΞ v,m,u " S v´mΞv,m,u and choose δ ě 0 such that µ´1 ą δ ą β´1. Using the semigroup smoothing property ( . ) we then have: Going from second to the third line we used that, by convexity of the integrand, the Riemann sum is bounded by the integral. In the last inequality we used that ş 1 0 r´β`δdr ă 8 since β´δ ă 1. Since δ is chosen so that 1´µ`δ ă 0, this is summable and yields desired bound ( . ).
It may appear a priori that we only have IΞ P C 2 , but a similar argument to [Gub , FH ] shows that actually ( . ) holds, which immediately implies thatδpIΞq " 0 as desired. The fact that IΞ P C γ 2 H α follows easily by taking β " 0 and noting that we then have }IΞ t,s } Hα À |Ξ| γ,α |t´s| γ`|δ Ξ| µ,α |t´s| µ . The continuity of I follows exactly as in [Gub , FH ] and is left to the reader.

Controlled Rough Paths according to the Semigroup
We now recall the notion of rough path introduced by Lyons in the 's (see for example [Lyo ] or [LQ ]). To treat SPDEs in Hilbert spaces, we could use an operator-valued definition of rough path as in [GT , BG ]. However, we will focus on equations driven by finite-dimensional Brownian motion and we would like to reuse already known results like Norris's Lemma or Malliavin calculus for rough paths of finite dimensions. We will therefore pursue a compromise and use the "classical" definition of a rough path for our driving noise, while slightly modifying the notion of a "controlled rough path" from [GT ] to encode the interaction of our class of integrands with the semigroup.

Definition . (Rough Path).
We say that a pair of functions pX, Xq P C 2 pR d qĈ 2 pR dˆd q satisfies Chen's relations if for all s ď u ď t: δX t,u,s " X t,s´Xt,u´Xu,s " 0 , δX t,u,s " X t,s´Xt,u´Xu,s " X t,u b X u,s .
For γ P p1{3, 1{2s and for two such pairs X " pX, Xq,X " pX,Xq we define the rough path metric ̺ γ as: Finally for γ P p1{3, 1{2s we define the space of rough paths C γ pr0, T s, R d q to be the completion with respect to ̺ γ of all smooth pairs pX, Xq P C 8 p∆ 2 , R d qĈ 8 p∆ 2 , R dˆd q satisfying ( . ).
For simplicity we write ̺ γ pXq :" ̺ γ p0, Xq. Note that the convergence with respect to above metric is implying the pointwise convergence thus Chen's relation is true for elements of C γ pr0, T s, R d q. Here instead of writing |X| γ,R d and |X| 2γ,R dˆd we made C R P S an abuse of notation by simply writing |X| γ and |X| 2γ and hope that no confusion will arise from this. The first equation in Chen's relation ( . ) actually tells us that X belongs to C 1 in a sense that we can write X t,s " δpX¨, 0 q t,s and so X is completely determined by X¨, 0 P C 1 . We decide not to use C 1 in the definition of the rough path since in our analysis we only care about the increments of functions and not about their precise value. Nevertheless we might sometimes neglect this and talk about X as a one time parameter function. One should think of X i,j t,s as postulating the value of the integral ş t s X i u,s dX j u which may not be defined classically through the theory of Young's integration [You ] since for that we need γ ą 1{2 in general. This motivates us to define a canonical lift of the smooth path to the rough paths and the definition of the geometric rough paths: ş t s δX u,s b dX u and the right hand side is a Riemann integral.
For γ P p1{3, 1{2s we say that the rough path which always holds for the canonical lift of a smooth path. We write C γ g pr0, T s, R d q Ă C γ pr0, T s, R d q for the subspace of geometric rough paths.
One can show that the space of geometric rough paths is the closure of all the smooth lifts with respect to the rough path metric ̺ γ .
Equations of interest to us are driven by . . , X d q is a rough path. We will typically use instead the shorthand notation F pu t qdX t , where we view F : H α Ñ LpR d , H β q. For simplicity we will denote the space LpR d , H β q by H d β . With this notation, our spaces of integrands are defined as follows.
Definition . . Let X P C γ pr0, T s, R d q for some γ P p1{3, 1{2s and let m P N. We say that pY, belongs to C 2γ 2 H m α . We then write pY, Y 1 q P D 2γ S,X pr0, T s, V q and define a seminorm on this space by: Similarly, its norm is given by Remark . . In the special case S " id, this is nothing but the usual notion of a controlled rough path introduced by Gubinelli in [Gub ] (see also [LY ] for a different perspective). In this case, we will omit the subscript S in the notations introduced above.
Note that we have the bound where the constant C depends only on γ and T and can be chosen uniform over T P p0, 1s. Given a controlled rough path according to S, we can define a corresponding 'stochastic convolution'.
exists as an element ofĈ γ H α and satisfies for every 0 ď β ă 3γ: Moreover the map S,X pr0, T s, H α q and one has the bound: Here the underlying constant depends on γ, d and T and is uniform over T P p0, 1s.
This follows from the definition of controlled rough path ( . ) and Chen's relation ( . ).
SinceδΞ v,m,u " S v´mΞv,m,u for someΞ satisfying ( . ) with µ " 3γ ą 1 and M " |R Y | 2γ,α |X| γ`} Y 1 } γ,α |X| 2γ , the existence of the limit in ( . ) and the bound ( . ) follow directly from ( . ). If we define Z t " ş t 0 S t´u Y u dX u then it is not hard to see thatδZ t,s " ş t s S t´u Y u dX u , so that ( . ) follows immediately from ( . ). We will address the continuity of integration map in Section . below.

. Composition with regular functions
Now, we need to restrict our study to a suitable class of non-linearities.

Definition . .
For some fixed α, β P R and k P N 0 we define the space C k α,β pH m , H n q as the space of k-differentiable functions G : H m θ Ñ H n θ`β for every θ ě α, and n, m P N 0 and such that D i G sends bounded subsets of H m θ to bounded sets of H n θ`β , for all i " 0, . . . k. For such functions }G} C k will represent some norm which depends on the first k derivatives and its exact form will be clear from the context. When m " n we will simply write C k α,β pH m q.
With this at hand: C R P S Lemma . . Let F P C 2 α,0 pH m , H n q with all derivatives up to order 2 bounded, let T ą 0 and pY, Y 1 q P D 2γ S,X pr0, T s, H m α q for some pX, Xq P C γ pr0, T s, R d q, γ P p1{3, 1{2s. Moreover assume that in addition Y P L 8 pr0, T s, H m α`2γ q and Y 1 P L 8 pr0, T s, H mˆd α`2γ q. Define pZ t , Z 1 t q " pF pY t q, DF pY t q˝Y 1 t q then pZ, Z 1 q P D 2γ S,X pr0, T s, H n α q and satisfies the bound: The constant C F depends on the bounds on F and its derivatives. It also depends on time T , but is uniform over T P p0, 1s.

Proof.
We only consider the case d " m " n " 1, the generalisation to higher dimensions being purely a matter of notations. From ( . ) since 0 ă γ ă 2γ ď 1 we have for any V P C 1 pr0, T sq and for any u P r0, T s and any α P R the following: Bound of Z 1 : First, we writeδZ 1 as pδZ 1 q t,s " pDF pY t qY 1 t´D F pY t qS t´s Y 1 s q`pDF pY t qS t´s Y 1 s´D F pY s qS t´s Y 1 s q pDF pY s qS t´s Y 1 s´D F pY s qY 1 s q`pDF pY s qY 1 s´St´s DF pY s qY 1 s q " I`II`III`IV .
Combining these bounds all together we obtain: Bound of R Z : The term VI can easily be bounded using bounds for II, III and IV: For V we have V "´F pY t q´F pS t´s Y s q´DF pY t qpY t´St´s Y s q¯`´F pS t´s Y s q´S t´s F pY s q¯ C R P S " VII`VIII.
By Taylor's theorem we obtain while the 'commutator' VIII is bounded by where we used ( . ) to go from the first to the second line. Combining both bounds on Z 1 and R Z we obtain the desired result.
. D 2γ,β,η S,X spaces Definition . . Let X P C γ pr0, T s, R d q for some γ P p1{3, 1{2s. Then for and β P R and η P r0, 1s define a space We also wroteĈ 0 " L 8 for η " 0 and, as usual, we will drop the subscript S when S " id. Note that by Lemma . , composition with regular functions maps D 2γ,2γ,η S,X pr0, T s, H α q to D 2γ,2γ,0 S,X pr0, T s, H α q for every η P r0, 1s. For simplicity, we also introduce the useful notation D 2γ X pr0, T s, H α q :" D 2γ,2γ,γ S,X pr0, T s, H α´2γ q .
Warning: we have shifted the space regularity in the definition of D 2γ X by 2γ in the right hand side. We will later solve our equations in the space D 2γ pr0, T s, Hq (α " 0).

Combining this with
we conclude that the remainder R Y t,s`S t´s Y s´Ys`p S t´s Y 1 s´Y 1 s qX t,s is of regularity |t´s| 2ε . We can similarly show that Y 1 P C ε and therefore D 2ε,2γ,0 X pr0, T s, H α q Ď D 2ε,2γ,0 S,X pr0, T s, H α q. The proof of the converse implication is analogous.
For the next proposition we use that the inner product on H extends uniquely to a continuous bilinear map x¨,¨y : H´αˆH α Ñ R for every α ě 0. pr0, T s, H´2 γ q and ψ P H 2γ , one has pxY, ψy, xY 1 , ψyq P D 2γ X pr0, T s, Rq. Also for any fixed t ď T we get a controlled rough path pxS t´¨Y , ψy, xS t´¨Y 1 , ψyq P D 2γ X pr0, ts, Rq. Moreover for fixed t ą 0 Similar bound holds for the other two controlled rough paths stated in the proposition, but with }h} replaced by }ψ} H2γ . Proof. The fact that the first two functions are controlled rough paths follows easily from Proposition . . For the third one we cannot use Cauchy-Schwarz straight away because h is not regular enough. Instead, we use ( . ) and write Since }Y } 8,H and }Y 1 } 8,H d 0 are finite we see that the last two terms of R Z are bounded by |v´u|p}Y } 8,H`} Y 1 } 8,H d 0 q}h}. For the first term we have:ˇˇż where we have used that 2γ ă 1. One similarly shows that |Z 1 | γ ă 8.
We finish this subsection by extending Lemma . to functions that lose some space regularity. Since the proof is identical to that of Lemma . , we omit it. Lemma . . Let σ ě 0 and F P C 2 α,´σ pH, H d q with all respective derivatives bounded. Let T ą 0 and pY, Y 1 q P D 2γ,2γ,0 S,X pr0, T s, H α q for some pX, Xq P C γ pr0, T s, R d q, γ P p1{3, 1{2s. Then pZ t , Z 1 t q " pF pY t q, DF pY t q˝Y 1 t q P D 2γ,2γ,0 S,X pr0, T s, H d α´σ q and one has }pZ, Z 1 q} X,2γ,α´σ À F p1`|X| γ q 2 p1`}Y, Y } D 2γ,2γ,0 S,X q 2 .
C R P S

. Stability of integration and composition
First we will give a meaning to the "distance" between two controlled paths that are controlled by two different rough paths. Then with the notion of these two distances we will state the continuity of two maps: integration and composition.
Definition . . For pY, Y 1 q P D 2γ S,X pr0, T s, H m α q and pV, V 1 q P D 2γ S,X pr0, T s, H m α q define a distance: We also measure the distance between two functions pY, Y 1 q P D 2γ,β,η S,X pr0, T s, H m α q and pV, V 1 q P D 2γ,β,η S,X pr0, T s, H m α q with: We make an abuse of notation by not writing dependence of d X,X,2γ and d 2γ,β,η on Y 1 and V 1 .
For the next two lemmas we are going to assume with X,X, Y, V as above that there exists M ą 0 such that |X| γ , |X| 2γ , }pY, Y 1 q} D 2ε,2γ,η S,X ă M and the same is true forX and V . We are not presenting the proofs of the following stability results since the ideas are exactly the same as in the proofs of their analogues Theorems . and . from [FH ]. The modifications needed for our case only involve replacing the sewing lemma by Lemma . and exploiting the fact that the regularity assumptions on F yield control on commutators of the type F pS t´s Y s q´S t´s F pY s q. These modifications were already used to similar effect in the proofs of Theorem . and Lemma . .
Consider pY, Y 1 q P D 2ε,2γ,0 S,X pr0, T s, H d α q and pV, V 1 q P D 2ε,2γ,0 S,X pr0, T s, H d α q that both satisfy the bounds with respect to M as above. Define pZ, Z 1 q :"´ż0 S¨´uY u dX u , Y¯, and similarly pW, W 1 q as a rough integral of pV, V 1 q. Then the following local Lipschitz estimates are true: with the underlying T -dependent constants uniform for T ď 1.
It may look like we are far from obtaining the stability result in the same Hölder regularity as our rough path X, but here ε can be taken arbitrarily close to γ which itself allows to take η arbitrarily close to ε. Note also that inequality ( . ) is true in spaces D 2ε S,X and not just in D 2ε,2γ,ε S,X . Lemma . . Let X,X, 1{3 ă ε ď γ ď 1{2 and η P r0, 1s. Let pY, Y 1 q P D 2ε,2γ,η S,X pr0, T s, H α q and pV, V 1 q P D 2ε,2γ,η S,X pr0, T s, H α q satisfy the bounds with respect to M as above. Let σ ě 0 and F P C 3 α,´σ pH, H d q. Define pZ, Z 1 q :" pF pY q, DF pY q˝Y 1 q and pW, W 1 q :" pF pV q, DF pV q˝V 1 q.
Then the following local Lipschitz estimates are true: Here d 2ε,2γ,0 pZ, W q contains H α´σ and H α`2γ´σ spatial norms and d 2ε,2γ,η pY, V q contains H α and H α`2γ spatial norms.

Rough PDEs
We now use the results obtained in the previous section to solve RPDEs in the Hilbert space H. First we consider equations without non-linear drift of the type Here L is as above, F is a C 3 function on H and X " pX, Xq P C γ pR`, R d q (meaning |X| γ,rS,T s and |X| 2γ,rS,T s are finite for all intervals rS, T s).
We will show that Lemma . and Theorem . guarantee that if pY, Y 1 q P D 2γ X pr0, T s, Hq, then yields again an element of D 2γ X pr0, T s, Hq. We now show that for T small enough this map has a unique fixed point:

Theorem . (Rough Evolution Equation)
. Given ξ P H, F P C 3 2γ,0 pH, H d q and X " pX, Xq P C γ pR`, R d q, there exists τ ą 0 and a unique element pY, Y 1 q P D 2γ X pr0, τ q, Hq such that Y 1 " F pY q and Proof. First note X " pX, Xq P C γ Ă C ε for 1{3 ă ε ă γ ď 1{2. Let T ă 1 we will find a solution pY, Y 1 q P D 2ε X pr0, T s, H 2ε´2γ q as a fixed point of the map M T given by ( . ). Then in the end we will briefly describe that one can actually make an improvement and show that pY, Y 1 q P D 2γ X pr0, T s, Hq. The proof is analogous to [FH , Thm . ], the only difference being that we have two different scales of space regularity for which we need to be able to obtain the bound ( . ). We will therefore show only invariance of the solution map ( . ), because proving it already contains all the techniques that are not present in the [FH , Thm . ].
Any semi-norm }¨} X,2ε will be taken in the H´2 γ space so sometimes we won't indicate this. Note that if pY, Y 1 q is such that pY 0 , Y 1 0 q " pξ, F pξqq then the same is true for M T pY, Y 1 q. We can therefore view M T as a map on the complete metric space: This is also true for the closed unit ball B T centred at t Ñ pS t ξ`S t F pξqX t0 , S t F pξqq P D 2ε X pr0, T s, H 2ε´2γ q. One can show using }pS¨ξ`S¨F pξqX¨0, S¨F pξqq} X,2ε,´2γ " 0 (since thatδpS¨ξq t,s " 0) that in fact: Note that by the triangle inequality for pY, Y 1 q P B T we have }pY, Y 1 q} D 2ε X À p1`}ξ}`}F pξq}qp1`|X| γ q. It remains to show that for T small enough M T leaves B T invariant and is contracting there, so that the claim follows from the Banach fixed point theorem. Constants below denoted by C may change from line to line and depend on γ, ε, X, X and ξ without mentioning. Nevertheless they are uniform in T P p0, 1s. Without loss of generality we assume that F is C 3 b , since by definition of C 3 2γ,0 pH, H d q function F sends bounded sets to bounded sets, which is the case for us since for pY, Y 1 q P B T , both |Y | 8,2ε´2γ and |Y 1 | 8,2ε´2γ are uniformly bounded by a constant depending on ξ. For pZ t , Z 1 t q " pF pY t q, DF pY t q˝Y 1 t q we have by Lemma . ( . ): Since pY, Y 1 q P B T , we obtain from ( . ) that }Y } ε,´2γ ď p|X| γ`1 qT γ´ε . One can also show along the same lines as in Lemma . that Therefore since T ă 1 we conclude that }Z} ε,´2γ À C F,ξ T γ´ε , where C F,ξ is a constant that also depends on initial condition. To estimate }M T pY q´S¨F pξqX¨, 0 } ε,2ε´2γ we useδpS¨F pξqX¨, 0 q t,s " S t F pξqX t,s and since 2ε ă 1 we can use a better bound from ( . ) to deduce: Finally we estimate the term }M T pY q 1 t´S t F pξq} H2ε´2γ : Putting it all together we can get that If T is small enough we guarantee that the left hand side of the above expression is smaller than 1, thus proving that B T is invariant under M T . In order to show contractivity of M T , one can use analogous steps to first show This guarantees contractivity for small enough T , completing the fixed point argument and thus showing the existence of the unique maximal solution to ( . ).
Let now pY, Y 1 q P D 2ε X pr0, T s, H 2ε´2γ q be the solution constructed above, we sketch an argument showing that in fact it belongs to D 2γ X pr0, T s, Hq. We know that Here R t,s " ş t s S t´r F pY r qdX r´St´s F pY s qX t,s´St´s DF pY s qF pY s qX t,s . From the estimate on R t,0 using ( . ) and since ξ P H, we see that ( . ) S,X pr0, T s, H´2 γ q (using again ( . )) and pF pY q, DF pY qF pY qq P D 2γ S,X pr0, T s, H´2 γ q which enables us to get an estimate for every β ă 3γ: }R t,s } H α`β À X }F pY q, DF pY qF pY q} X,2γ |t´s| 3γ´β .
Taking β " 2γ and using ( . ) again we show that Y PĈ γ pr0, T s, Hq, which completes the proof that pY, Y 1 q P D 2γ X pr0, T s, Hq.
For N satisfying the same assumptions as the nonlinearities F i in Theorem . , we immediately get local solutions to equations of the type for the rough path X t " pX 1 t , . . . , X d t q P C γ pr0, T s, R d q for γ P p1{3, 1{2s. This is because we can simply treat this equation as driven by the rough pathX t " pX 1 t , . . . , X d t , tq. However, we can do a bit better than that and obtain weaker assumptions on N .
Definition . . Let k, n P N 0 , we call a function N P C k α,β pHq to be of polynomial type n and write N P Poly k,n α,β pHq if for all σ ě α and 0 ď i ď k there exists C σ,i ą 0 such that for all x, y P H σ Theorem . (Rough Nonlinear PDE). Let γ P p1{3, 1{2s and X " pX, Xq P C γ pR`, R d q. Then, given ξ P H, F P C 3 2γ,0 pH, H d q, and N P Poly 0,ń κ,´δ pHq for some n ě 1, some 1´δ ą γ and some small κ ą 0, there exists τ ą 0 a unique element pu, u 1 q P D 2γ X pr0, τ q, Hq such that u 1 " F puq and We call such u t a mild solution to the Rough PDE: Proof. The proof is almost identical to that of Theorem . once we can deal with the non-linearity N . First we take ε P p1{3, 1{2s so that ε ă γ hence 1´δ´ε ą 0 by our assumption on δ. For pu, u 1 q P B T , we show that V t " ş t 0 S t´r N pu r qdr P D 2ε X . This is possible if we take V 1 " 0 and thus R V "δV . Since the assumption 1´δ ą γ implies δ ă 2{3 it is possible to find β ą 0 such that β ď δ and 1´2ε ą δ´β whence We have used above that β ď δ ă 2{3 ď 2ε and hence: Here we need to further impose β´2γ ě´κ (which is possible by an appropriate choice of β) so that we can evaluate N puq for u P H β´2γ . Similarly to above we get The last inequality serves two roles: First, since ε ă γ can be taken arbitrarily close to γ, it follows from 1´γ´δ ą 0 that for some σ ą 0 we have Together with the invariance estimates established in the proof of Theorem . , we conclude that the fixed point map M T leaves B T invariant for sufficiently small T . This bound also shows that } ş¨0 S¨´rN pu r qdr} γ ă 8 which is needed to prove that this solution actually lives in D 2γ X . The contractivity of M T is obtained in a similar way, now using the local Lipschitz property of N . Remark . . Assumption N P Poly 0,ń κ,´δ leads to a small problem when we want for instance to take H " L 2 because then N is nonlinear function that needs to act on the space of distributions H´κ. One can actually remove this problem and show the existence of the solution in D 2γ X for N P Poly 0,n 0,´δ . This can be achieved by first solving the equation in the spaces D 2ε S,X pr0, T s, H´2 γ q X`Ĉ η pr0, T s, HqˆL 8 pr0, T s, H d 2ε´2γ q˘, for some η ă ε ă γ and then again show that all the regularities can be improved and that the solution is indeed in D 2γ X pr0, T s, Hq. We decided to avoid this and not to use even more norms on the different space time scales for simplicity.
" D 2γ,2γ,0 X and since both integration and composition with smooth functions preserves D 2ε,2γ,0 X , one might ask why not to solve these equations in D 2γ,2γ,0 X or even in D 2γ X in the first place. First if we would solve our equations in D 2γ X pr0, T s, Hq with initial condition in H then we will run into problem of estimating the term }S t ξ´S s ξ} H . This term would have to be bounded by |t´s| 2γ which is not true for general ξ P H but true for ξ P H 2γ . This suggests that one must look for the solution in the space like D 2γ,2γ,0 X . We believe that this indeed can be done. This approach would have an advantage that estimates on the composition with the regular function in space D 2γ,2γ,0 X automatically follows from the usual estimate on the control rough paths. Nevertheless we decided to stick to the space D 2γ,2γ,0 S,X because the operatorδ acts nicely on the integrals of the form ş t 0 S t´s Y s dX s . Otherwise we would always have to deal with estimating two kinds of expressions: ş t s S t´r F pu r qdX r and ş s 0 pS t´r´Ss´r qF pu r qdX r . In conclusion, it seems that working in spaces D 2γ,2γ,0 S,X and D 2γ,2γ,0 X is essentially equivalent but in one space it is easier to estimate integrals and in the other it is easier to estimate composition with the functions.

. Continuity of the solution map
In this subsection we are going to use stability results for integration and composition in order to prove continuity of the solution map of the RPDEs (which in the classic literature for solutions of RDE's is called Itô-Lyons map).

Theorem . (Stability of solution to RPDE).
Let γ P p1{3, 1{2s and X,X P C γ . Let ξ,ξ P H, let F P C 3 2γ,0 pH, H d q, and N P Poly 0,n 0,´δ pHq for some n ě 1 is a function of polynomial type for some 1´δ ą γ. Define pu, F puqq P D 2γ X pr0, τ 1 q, Hq to be a maximal solution to the RPDE: and pv, F pvqq P D 2γ S,X pr0, τ 2 q, Hq to be a maximal solution of the same RPDE but driven by the rough pathX and initial conditionξ. Assume that ̺ γ pXq " |X| γ`| X| 2γ ă M and }ξ} ă M and same withX andξ. Then for every 1{3 ă ε ă γ and 0 ď η ă 3ε´2γ there exists time T ă 1^τ 1^τ2 such that for the following seminorm taken with respect to this time T we have: Moreover if both solutions are global (i.e. τ 1 " τ 2 " 8) then ( . ) holds for all T ą 0.
Proof. Note that continuity of the solution is proven in a bit worse Hölder regularity, but the space regularity remains the same. Moreover case when η " 0 is immediate by the if we prove the case for η ą 0 simply because d 2ε,2γ,0 pu, vq À T,η }ξ´ξ}`d 2ε,2γ,η pu, vq. First we will take T small enough such that both solutions u and v satisfy for some The fact that such T exists was shown in the proof of invariance in Theorems . and . . This guarantees that all bounds with respect to M in Lemma . and Lemma . are satisfied and moreover the right hand side of inequality ( . ) is independent of solutions u and v. From now on we will use without further mentioning that }F pξq´F pξq} À F }ξ´ξ}.
Recall that R U "δU and R V "δV and V 1 " U 1 " 0. Since N is locally Lipschitz then similarly as in Theorem . we can show that we can pick a β ą 0 such that 1`β´δ´2ε ą 0 and: for some σ ą 0. In the last step we have used inequality We have used above Lemma . for the first inequality and Lemma . for the second inequality. To deal with the term }Ξ´Ξ} 8 we use to finally deduce that for some potentially even smaller σ ą 0: Now pu, F puqq is a fixed point of the map: and similarly for pv, F pvqq withX. Putting the bounds on d 2ε,2γ,η pZ, W q and on d 2ε,2γ,η pU, V q together we get: If we take T " τ pM, F, ε, γ, ηq sufficiently small such that C M T σ ď 1{2 then we get: Now if we know that both solutions pu, F puqq and pv, F pvqq are global in time we can iterate stability result . in order to obtain it for an arbitrary T ą 0. This can be done by investigating more carefully the proof of Theorems . and . and observing that the inverse of time T 0 within which invariance and contraction holds bounded from above by some powers of ̺ γ pXq and }ξ}. This then allows to show that we can bound from above the number of times we would need to iterate . to get to the time T .
Next we state that for every global solution u with the noise X and initial condition ξ there is a small ball around X and small ball around ξ such that for every noiseX and initial conditionξ inside these balls the size of solution v " vpX, ξq is not much bigger than the size of the solution u. Proof of the following proposition is quite standard and again uses the iteration of . .
Constant C " CpM, R, T q is locally bounded function in all three variables.

. Solutions to SPDEs
First of all we define the spaces of controlled rough paths that are allowed to blow up in finite time. For every Banach space V define a new Banach spaceV " V \ t8u.
The topology on this space is induced by the basis containing open balls of V and the sets of the form tv P V : }v} V ě N u \ t8u for every N ą 0. Using this we define the space of controlled rough paths that might blow up in finite time: The τ in the above definition is denoting a blow up time of pu, u 1 q and can be taken 8 for the controlled rough paths which have finite D 2γ,β,0 S,X norm on every compact interval.
All our analysis was purely deterministic so far. There is a wide class of Gaussian processes that can be lifted almost surely to a rough path and thus our theory is giving a pathwise notion of solution for such SPDEs driven by these Gaussian processes. Equations that we are going to investigate are driven by Brownian motion and we now briefly recall how one defines a Brownian rough path. The following definition requires a proof, which can be found in [FH ].
ş t s δB r,s b dB r as an Itô integral. Then @γ P p1{3, 1{2q and T ą 0 for a.e. ω P Ω One would like to know that the rough integrals defined earlier against these Brownian lifts coincide with Itô (resp. Stratonovich) integrals for a suitable class of integrands: Proposition . . Let pB t q tě0 : Ω Ñ R d be a d-dimensional Brownian motion defined on the filtered probability space pΩ, pF t q tě0 , Pq and let pY, Y 1 q PD 2ε,2γ,0 S,B pR`, H´2 γ q be such that pY, Y 1 q is adapted to the filtration pF t q tě0 and such that, for every L ą 0 there exists a stopping time T L andL ą 0 such that }Y t }`}Y 1 t } ďL almost surely for t ď T L . For L ą 0 and t ą 0, set where the integral is an Itô integral. Then the process Z L has a continuous version (still denoted by Z L ) such that for any random time tpωq with 0 ď tpωq ă T L pωq, the following equality with the rough integral holds almost surely: since T L is a stopping time, the integrand is adapted to the Brownian filtration. Let P n " ts n k u 8 k"0 be a sequence of increasing countable subsets of R`such that Ť n P n is dense in R`and s n k ă s n k`1 for all n, k P N. Denote by π n " trs n k , s n k`1 s : s n k P P n , k P Nu the sequence of partitions formed from P n and |π n | " sup kě1 t|s n k`1´s n k |u is the size of partition. It follows that Z L t is defined as a limit in probability: We can now extract a subsequence of partitions (which we still denote P n ) such that the above limit holds almost surely. On the other hand, since pY, Y 1 q PD 2ε,2γ,0 exists and one can verify that it is equal to: We can therefore easily see that for every t ą 0 the L 2 pΩq norm of the difference of these two integrals is:ˇˇZ We will show now that the right hand side is zero. Define a (discrete time) martingale started at M n 0 " 0 and with increments M n k`1´M n We use the fact that all the infinite sums above are finite because of the presence of the indicator function. Moreover the last inequality is true because the Brownian scaling gives |B Itô v,u | 2 L 2 pΩq À |v´u| 2 . Since Ť n P n is dense in R`we have |π n | Ñ 0 as n Ñ 8. Therefore by Fatou's lemma right hand side of ( . ) is indeed zero thus showing that for all t ą 0 we have almost surely Z L t "Z L t . Now one can choose a continuous version of the Itô integral Z L which is still equal almost surely toZ L t for every t ą 0. We can therefore evaluate Z L t at a random time 0 ď tpωq ă T L pωq to deduce that the identity holds almost surely.
Before will now formalize the notion of a local in time solution for an Itô SPDE.

Definition . . Let pB t q tě0
: Ω Ñ R d be a d-dimensional Brownian motion defined on the filtered probability space pΩ, pF t q tě0 , Pq. Let ξ P H, δ P r0, 1q and consider locally Lipschitz continuous maps N : is a continuous stochastic process u together with the stopping time τ such that almost surely on the event tt ď τ u, u t satisfies where the last integral is taken in the sense of Itô. We furthermore impose that there exists L ą 0 such that sup 0ďtďτ }u t } ď L almost surely.
(ii) We say pu, τ q is a maximal mild solution of ( . ) if lim tÑτ }u t } " 8 almost surely and there exists a sequence of local mild solutions pu n , τ n q with increasing τ n such that lim nÑ8 τ n " τ almost surely and u n t " u t almost surely on tt ă τ n u.

Theorem . . Let ξ P H and functions F and N be as in Theorem . . Let
Then there exist random blow up times τ 1 , τ 2 ą 0 and controlled rough paths pu, F puqq P D 2γ B pr0, τ 1 q, Hq, pv, F pvqq P D 2γ B pr0, τ 2 q, Hq such that they are almost surely maximal solutions of ( . ) with X replaced by B Itô and B Strat respectively.
In addition the above solutions are adapted processes when viewed as elements of D 2ε,2γ,0 S,B`R`, H´2 γ˘. As a consequence the following holds: (i) pu, τ 1 q is a maximal mild solution to the Itô SPDE: (ii) pv, τ 2 q is a maximal mild solution to the Itô SPDE: Proof. We first show the result for pu, F puqq. Local solution theory for ( . ) with X replaced by almost every realization of B Itô is provided by Theorem . . The fact that τ 1 is a stopping time is easy to verify. Note that the map Bae r0,ts Þ Ñ pB, B Itô qae r0,ts P C γ pr0, ts, R d q is measurable. For almost every ω and every t ă τ 1 pωq, the solution pu, F puqq P D 2ε,2γ,0 S,B`r 0, ts, H´2 γ˘t o ( . ) is a continuous image of the noise pB, B Itô qae r0,ts . Viewing pu t pωq, F pu t pωqqq as an element ofD 2ε,2γ,0 S,Bpωq`R`, H´2 γ˘w e deduce that is adapted to σpB s,r , B Itô s,r : 0 ď r ď s ď tq " σpB s : 0 ď s ď tq " F t . Let L ą 0 and define a stopping time T L " inftt : }u t } ě Lu then the local boundedness of F implies that there existsL ą 0 such that almost surely for t ă T L : For t ą 0 define the process u L t as: where the existence of the Itô integral is guaranteed by an almost sure bound ( . ). By definition of our notion of solution to ( . ), we furthermore know that, for any (random) time tpωq ă T L pωq, one has the identity By Proposition . and equation ( . ), we conclude that, almost surely, u tpωq pωq " u L tpωq pωq, provided that we consider a continuous version of u L . This also shows that pu L , T L q is a local mild solution of the Itô SPDE ( . ). Whenever }u t } is finite we can always restart the equation ( . ) (with X replaced by B Itô ) with initial condition u t and extend the solution further in time therefore almost surely T L Ñ τ 1 as L Ñ 8. Moreover T L clearly increases as L increases and u L t " u t on tt ă T L u thus showing that pu, τ 1 q is indeed a maximal solution of ( . ).
Regarding the solution pv, F pvqq the proof is the same once we notice that and that all the above integrals make sense as elements of H. Then we apply Proposition . again for the rough integral with respect to B Itô in ( . ) ( . ) is true for all T ą 0.

. Malliavin differentiability and the Jacobian
In this subsection we show the Malliavin differentiability of the solutions to RPDEs driven by general Gaussian rough paths, using only that the solution does not blow up in finite time. Unfortunately the method is non constructive and only gives the knowledge that Malliavin derivative exists and lies in the Shigekawa-Sobolev space D 1,2 loc . In particular, it does not automatically imply that this Malliavin derivative is a controlled rough path itself and / or that it solves some RPDE. Nevertheless this is not so important for our analysis since our main result regarding the non-degeneracy of the Malliavin matrix does not require Malliavin differentiability per se. This is due to the fact that we will define Malliavin matrix only using the existence of linearisation of solution. Moreover if say equation of our interest is driven by the Brownian motion and we can show Malliavin differentiability then Malliavin derivative satisfies an SPDE and therefore almost surely the RPDE by Theorem . . This restriction to the Brownian case is also performed because for general rough paths the a priori bounds are not easy to obtain. Despite the fact that we will later simply assume the Malliavin differentiability and won't explicitly use Theorem . we still present it together with the proof as a result on its own. Before we proceed, we quickly recall the Cameron-Martin theory for general centred Gaussian rough paths.
Let Ω " Cpr0, T s, R d q and let X : Ωˆr0, T s Ñ R d be the a canonical centred Gaussian process so that X t pωq " ωptq. The Gaussian law of X is completely determined by its covariance function R X : r0, T s 2 Ñ R dˆd . For p ě 1 define the D p-variation of R on a rectangle IˆI 1 Ď r0, T s 2 to be: }R X } p,IˆI 1 :"´sup P PπpIq P 1 PπpI 1 q ÿ rs,tsPP rs 1 ,t 1 sPP 1 |ErδX t,s b δX t 1 ,s 1 s| p¯1 {p .
πpIq denotes here the partitions of I. Similarly one can define }R X i } p,IˆI 1 . The Cameron-Martin space CM T Ă Cpr0, T s, R d q is a Hilbert space which consists of the paths v t " ErZX t s for Z lying in the first Wiener chaos W 1 which is an L 2closure of spantX i t : t P r0, T s, 1 ď i ď du. See [FH , Chap. , ] for the description of the regularity of the Cameron Martin space and for precise conditions on the covariance function which guarantee that X can be lifted almost surely to a rough path in a canonical way. In the case when the Gaussian process is a Brownian motion, For γ P p1{3, 1{2q and a generic pX, Xq P C γ pr0, T s, R d q let h : r0, T s Ñ R d be sufficiently smooth, the translation operator of X in the direction v is defined by Here by sufficiently smooth we understand that all three integrals above make sense classically and moreover makes the operator T h a continuous map C γ to itself. In fact it is true for h P H 1 . Moreover if X is the Brownian rough path (either Itô or Stratonovich) we have that for almost every ω P Ω and every h P CM T we have: A similar result holds for a general Gaussian rough path with the regular enough covariance. Let X be a centred Gaussian rough path which almost surely lies in C γ with covariance R and let CM T be its Cameron Martin space. Assume that for some p P r1, 2q every v P CM T has finite p-variation }v} p-var,rs,ts over the interval rs, ts Ď r0, T s and it satisfies an inequality: Then if X almost surely satisfies equality ( . ) it is easy to show that, almost surely for every h P CM T , We The domain of this operator is denoted by D 1,2 . If we denote by F the σ-algebra of the Wiener space pΩ, CM, Pq then we say that Y P D 1,2 loc if there exists a sequence pΩ n , Y n q ně1 Ď FˆD 1,2 such that Ω n Ò Ω and Y " Y n almost surely on Ω n . See the book [Nua ] for an introduction to Malliavin Calculus. Having all this at hand we are ready to present a general statement on the Malliavin differentiability of the solution to the Stochastic RPDE driven by quite general Gaussian rough path and given that this solution does not explode until some deterministic time T .

Theorem . (Malliavin differentiability).
Let pX t q tPr0,T s be a d-dimensional, continuous Gaussian process with independent components defined on the probability space pΩ, F , Pq. Let the covariance R of X be such that there exist M ă 8 and p P r1, 2q such that for i P t1, . . . , du and rs, ts Ď r0, T s, }R X i } p,rs,ts 2 ď M |t´s| 1{p .
Let γ P p 1 3 , 1 2p q and for almost every ω let pupωq, F pupωqqq P D 2γ Xpwq pr0, τ pωqq, Hq be a local mild solution to the Stochastic RPDE: du t pwq " Lu t pwqdt`N pu t pwqqdt`F pu t pwqqdX t pwq, u 0 " ξ P H, such that }u} 8,r0,T s ă 8 almost surely. Then for all 0 ď t ď T the solution u t is Malliavin differentiable and u t P D 1,2 loc . Proof. Fix γ P p 1 3 , 1 2p q, assumptions on the covariance R guarantee (see [FH , Chap. ]) that there is a canonical lift of X to a rough path in C γ pr0, T s, R d q and that, for every h P CM T , }h} p-var,rs,ts ď }h} CM T |t´s| 1{2p .
Moreover for almost every ω P Ω and every h P CM T we have where T h denotes the translation operator by h which is well-defined thanks to ( . ) (see Theorem . , Proposition . and Theorem . in [FH ]). Because of this last property we will not distinguish between T h pXpωqq and Xpω`hq and from now on we will simply write ω`h to denote any of them, we also abuse the notation and simply write Xpωq " ω. Fix t P r0, T s and define an event B n " tω : }upωq} 8,r0,T s ď n{2, ̺ γ pωq ď nu.
Let ω P B n then from Proposition . we know that for such ω there exists a small number σ n (independent of ω and only dependent on n, ξ and the equation itself) such that for all h P Ω with ̺ γ pω`h, ωq ď σ n we have }upω`hq} 8,r0,T s ď 2}upωq} 8,r0,T s . Assume that }h} CM T ď σ 1 n :" σn 2n^1 then by ( . ) we have: This is showing that for the event A n " tω : sup }h}CM T ďσ 1 n }u t pω`hq} ď n, ̺ γ pωq ď nu, we have B n Ď A n and since }u} 8,r0,T s ă 8 almost surely we have that B n Ò Ω thus implying A n Ò Ω. It is also possible to find a σ-compact set G n Ă A n such that A n {G n is a null set. For A P F we define: Then define u n t :" φpρ Gn {σ 1 n qu t for φ P C 8 c pRq non-negative function such that |φptq| ď 1 and |φ 1 ptq| ď 4 for @t, φptq " 1 for |t| ď 1{3 and φptq " 0 for |t| ě 2{3. Then we have that u n t " u t on G n and }u n t } ď ½ tρG n ď 2σ 1 n 3 u }u t } ď n.
For the first term we proceed exactly as in Proposition . . from [Nua ] to deduce that for }h} CMT ď σ 1 n {3 }u n t pω`hq´u n t pωq} À T,n }h} CM T . For the underlying constant which is deterministic and depends only on T, n, initial condition and the equation itself. Exercise . . in [Nua ] shows that such local Lipschitz continuity guarantees that pG n , u n t q is the localizing sequence required for the definition of D 1,2 loc . Assume now that F P C 8 2γ,0 pH, H d q, and N P Poly 8,n 0,´δ pHq for some n ě 1. We then show that the Jacobian of the solution is related to the Malliavin derivative for RPDEs in the same way as for SDEs. The unique mild RPDE solution to the equation ( . ) driven by the geometric rough path X P C γ g and passing through the point u s " ξ gives rise to the solution flow Φ ξ t,s pXq " pu t , u 1 t q up until the blow up time. More formally Φ .,s : HˆC γ g prs, T sq ÑD 2γ X prs, T s, Hq.
The spaceD 2γ X is defined similarly to the spaceD 2γ,β,0 S,X from the previous subsection and takes into account the fact that the solution may blow up before the time T . The derivative of the flow with respect to the starting point is called the Jacobian and is denoted by J X t,s . J X t,s ζ :" d dε Φ ξ`εζ t,s pXq. If X is a rough path lifted from a smooth path and we can show that the solution to ( . ) is global for every initial condition then from classical theory of PDEs one can show (say using implicit function theorem) that the Jacobian exists at every ζ P H and will satisfy linearised equation: dJ X t,s ζ " LJ X t,s ζdt`DN pu t qJ X t,s ζdt`DF pu t qJ X t,s ζdX t , J X s,s ζ " ζ. ( . ) Or in the mild form: J X t,s ζ " S t´s ζ`ż t s S r´s DN pu r qJ X r,s ζdr`ż t s S r´s DF pu r qJ X r,s ζdX r . ( . ) From this representation and from the fact that pu, u 1 q is controlled by X we deduce that if such J X t,s ζ satisfies this equation then it is also controlled by X. Consider also the directional derivative of the flow in the direction of the noise: For h sufficiently smooth. Once again if X is lifted from a smooth path and h is smooth then classical PDE theory is telling us that D h Φ ξ t,0 pXq exists and that due to variation of constants formula it satisfies: The same passes to the geometric rough path in the limit: Proposition . . Let γ P p1{3, 1{2s and X P C γ g pr0, T s, R d q. Assume that for F P C 8 2γ,0 pH, H d q, and N P Poly 8,n 0,´δ pHq solution to the equation ( . ) exists in D 2γ X pr0, T s, Hq for every initial condition ξ P H. Let h P C p-var pr0, T s, R d q with complementary Young regularity γ`1{p ą 1. Then for @ζ P H both J X t,s ζ D h Φ ξ t,0 pXq exist as elements of D 2γ X pr0, T s, Hq, J X t,s ζ satisfies RPDE ( . ) and this Duhamel's formula holds: where the right hand side is well-defined as a Young integral. Proof. We proceed like in the similar result for RDEs from [FH ]. Let X n " X c pX n q be canonical lift of a smooth path X n such that X n approximates X with sup n ̺ γ pX n q ď ̺ γ pXq. Then the RPDE solution pu n , F p 9 u n qq to the equation ( . ) driven by X n lies in D 2γ X n pr0, T s, Hq and converges to pu, F puqq in the d 2ε,2γ,η metric from the global continuity result of Theorem . . Now if we take a smooth approximation of h as well say h m then from above we know that D h m Φ ξ t,0 pX n q and J X n t,s ζ and equations ( . ) and ( . ) are satisfied for these smooth approximations. Passing to a limit as n and m go to infinity we obtain the desired result. H. Then, for any t ą 0, define the Malliavin matrix M t : H Ñ H by

Definition . . Assume that ( . ) has global solutions for every initial condition in
.

Smoothing property of the solution
Due to the smoothing properties of the semigroup we expect that the solution is going to have a better spatial regularity after some time. In fact we are going to show that if we start our equation from ξ P H then the solution immediately belongs to H β for every positive β.

Proof.
From the mild formula of solution we get that: then taking 0 ă ν ă γ we can deduce from the property of the semigroup and bounds on rough integration: }u t } Hν À |t´s|´ν}u s }`T 1´δ´σ p1`}u} 8,r0,T s q n`̺ γ pXq}pu, F puqq} D 2γ X pHq T γ´ν . From the similar arguments of iteration as in Theorem . we can get that there exist C 1 M which is locally bounded in M such that }pu, F puqq} D 2γ X pHq T γ´ν ă C 1 M therefore we indeed deduce that for all 0 ď s ă t ă T

R F T
where C M " ̺ γ pXqC 1 M`p 1`M q n and T σ " T 1´δ´ν^T γ´ν . Thus since both functions F and N act on H ν for ν ą 0 the same way as on H we can now solve our equation with this new initial condition u t P H ν which from ( . ) satisfies }u t } Hν ď Cpt´ν M`C M T σ q ": M t (here constant C is the constant that is discarded by the À sign). This gives us that there exists τ " τ pM t q ą 0 such that the solution map is invariant and a contraction on the space D 2γ X prt, t`τ s, H ν q (or rather on some ball in this space). Thus we get that pu, F puqq P D 2γ X prt, t`τ s, H ν q. Now from ( . ) we get that all 0 ď s ă t`τ }u t`τ } Hν À |t`τ´s|´νM`C M T σ .
Picking s " τ we get again }u t`τ } Hν ď M t and thus starting the equation from the initial condition u t`τ and since τ only depended on M t we can get the solution on D 2γ X prt`τ, t`2τ s, H ν q. Thus summarising we get pu, F puqq P D 2γ X prt, t`2τ s, H ν q with again }u t`2τ } Hν ď M t . Bootstrapping this further since τ ą 0 is fixed we can get to time T in finite number of these iterations and indeed get pu, F puqq P D 2γ X prt, T s, H ν q.
In order to prove this proposition for arbitrary β ą ν ą 0 denote t 0 " tβ{ν and M 0 " M . Without loss of generality let β{η be an integer. Denote by M 1 " }u} 8,ν,rt0,T s thus from ( . ) and we have our solution pu, F puqq P D 2γ X prt 0 , T s, H ν q. Therefore proceeding exactly the same like in the beginning of the proof we get for all t 0 ď s ă r }u r } H2ν À |r´s|´νM 1`CM1 T σ .
As a consequence we get the solution pu, F puqq P D 2γ X pr2t 0 , T s, H 2ν q with M 2 " }u} 8,2ν,r2t0,T s . Recursively we get for every n such that nt 0 ă T , pu, u 1 q P D 2γ X prnt 0 , T s, H nν q and with M n :" }u} 8,nν,rnt0,T s we have the recursive inequality where we took into account that t " t 0 ν{β. Solving this recursive inequality we get that for some other constants σ " σpnq and C M " CpM, nq we have M n À t´n ν M`C M T σ . Picking n " β{ν we indeed get the result.
Note that we actually use in the proof that time interval on which the solution map is contractive and invariant also depends on the norms of F and N which can be different when acting on H ν for different values of ν. But since we want to make an improvement of space regularity only by the finite amount β we can pick the largest value of the norms of F and N only up to their action on H β . This smoothing property is going to help us to overcome the issue that solution pu, F puqq P D 2γ X pr0, T s, Hq to an RPDE lives in a space H´2 γ as a controlled rough path while just as a function of time u t P H. This makes it difficult to investigate the properties of u t in the Hilbert space H since in this space we cannot make an advantage of the fact that u is actually a controlled rough path.

Rough Fubini Theorem
In this section we will only work with the usual notion of the controlled rough path. We consider a wide class of processes Y t,s which are controlled rough paths in both of R F T their time directions. For such processes double rough integral can be defined and we will show when the order of integration can be swapped.
Definition . . Let γ P p1{3, 1{2s and X P C γ . We say that the process Y : r0, T sr 0, T s Ñ R dˆd is jointly controlled by X and write Y P D 2γ 2,X pr0, T s 2 , R dˆd q if for every fixed s P r0, T s we have that Y¨, s and Y s,¨a re both controlled rough paths with respect to X. In this case we write: Moreover we require for every fixed s P r0, T s that both Y 1 s,¨a nd Y 2 ,s are controlled rough paths such that Y 1,2 " Y 2,1 , where we write: Note that this also ensures that for every fixed 0 ď u ď v ď T both R 1 v,u p¨q and R 2 v,u p¨q are controlled rough paths. This can be shown by verifying this nice formula: Denote any side of this equality by Rpt, s, v, uq. We call Y 1 , Y 2 first order Gubinelli derivatives and Y 1,2 second order Gubinelli derivatives. Strictly speaking, the whole tuple pY, Y 1 , Y 2 , Y 1,2 q is an element of D 2γ 2,X since Y 1 , Y 2 , Y 1,2 need not be unique. We introduce the following seminorms: for any function Z : r0, T s 2 Ñ R dˆd }Z} 8,γ " sup 0ďsďT sup 0ďuăvďT |Z s,v´Zs,u | |v´u| γ ; }Z} γ,8 " sup 0ďuďT sup 0ďsătďT |Z t,u´Zs,u | |t´s| γ .

R F T
The total norm of the remainder is then defined as Finally putting it all together we can define a seminorms on D 2γ 2,X pr0, T s 2 , R dˆd q by: The following lemma about properties of the above seminorms is easy to verify.

Example .
Let K and H be Banach spaces and V P D 2γ X pr0, T s 2 , Kq, Z P D 2γ X pr0, T s 2 , Hq. Let B : KˆH Ñ R dˆd be a bilinear map. Then defining Y u,s :" BpV u , Z s q we have Y P D 2γ 2,X pr0, T s 2 , R dˆd q. Moreover with the abuse of notation when B act on K b R d or H b R d component wise: Later we will see a more sophisticated example where one cannot split Y so easily into the inner product of two controlled rough paths.
First we will show that integrating the jointly controlled rough path along one of the directions is creating a usual one time variable controlled rough path.
2,X pr0, T s, R dˆd q. Then writing pV r , V 1 r q :"´ż r 0 Y r,s dX s , Y r,r`ż r 0 Y 1 r,s dX s¯, pZ r , Z 1 r q :"´ż t r Y s,r dX s ,´Y r,r`ż t r Y 2 s,r dX s¯, defines controlled rough paths V P D 2γ X pr0, T s, R d q and Z P D 2γ X pr0, ts, R d q.

Proof.
A straightforward computation shows that: Using assumptions on the uniform norms on Y 1 and R 1 as well as bounds on the rough integrals one can indeed show that |R V v,u | À Y,T |v´u| 2γ and |δV 1 v,u | À Y,T |v´u| γ . For Z we have: One can easily show that |R Z v,u | À Y,t |v´u| 2γ and |δZ 1 v,u | À Y,t |v´u| γ .
Before we proceed we need a good notion of a smooth approximation of the jointly controlled rough path with respect to the smooth approximation of the rough path. We refer the reader to the paper [HK ] where the authors showed that where C 2γ Ă C 2γ is a closure of smooth functions with respect to the 2γ-Hölder norm. This means that for every X " pX, Xq P C γ pr0, T s, R d q there exists a unique X g " pX, X g q P C γ g pr0, T s, R d q and a unique f P C 2γ pr0, T s, R dˆd q with f 0 " 0 such that X t,s " X g t,s`δ f t,s .
Having this decomposition one can show that for pY, Y 1 q P D 2γ X pr0, T s, R d q the following integral formula holds: and the second integral on the right hand side makes perfect sense as a Young's integral, where we understand the product Y 1 s¨d f s of two dˆd matrices as a Frobenius inner product: for A, B P R dˆd set A¨B " trpA T Bq.
2,X pr0, T s 2 , R dˆd q admits a smooth approximation if there exist sequences X n P C 8 pr0, T s, R d q, f n P C 8 pr0, T s, R dˆd q and Y n P D 2γ 2,X n pr0, T s 2 , R dˆd q such that for X n " X c pX n qp 0, δf n q the following approximations hold: }Y 1´Y 1 n } γ`} R Y´RY,n } 2γ " 0 .

R F T
An example of Y that admits a smooth approximation can be Y from Example since both V and Z are the usual controlled rough paths by X and can be each smoothly approximated.

Remark . .
We sketch an argument on why a classical (one time variable) controlled rough path V can always be smoothly approximated. To see this, one first shows that the equality δV t,s " V 1 s X t,s`R V t,s is equivalent to showing that V " V 1 ă X`U for some U P C 2γ and ă denotes some sort of the paraproduct (which is a continuous bilinear map, see [GIP ] for the definition and properties of the Bony paraproduct on the Fourier space). This paraproduct ă can for example be defined by where ψ n,x is an L 2 -normalised wavelet basis. Then one will simply take a smooth approximation of X say X n and define V 1 n and U n by mollifying V 1 and U . Then defining V n :" V 1 n ă X n`U n from continuity of the paraproduct ă one can then show that pV n , V 1 n q converges to pV, V 1 q in the rough path metric. Though this smooth approximation of the classical rough path is not canonical we will see later on that for our purposes we will only need an existence of some smooth approximation and we do not care which particular one is it.
Since for Y P D 2γ 2,X pr0, T s 2 , R dˆd q we can integrate with respect to X in both of its time directions, a natural question is if the order of integration matters. Even though we believe that Fubini like theorem holds for every Y P D 2γ 2,X pr0, T s 2 , R dˆd q we only show the proof for Y that can be smoothly approximated. For our purposes this is going to be sufficient since we will apply these results later to a case similar to Example . First we show how to swap the order of integration where the limits of the second integral are time variables that are also integrated. It turns out that unlike for usual integration, there is in general a correction term appearing for non-geometric rough paths. For the purpose of clarity we use in this section notation for the rough integral using the bold letter dX, reserving dX for Young integrals.
Theorem . . Let γ P p1{3, 1{2s, X P C γ pr0, T s, R d q and let Y P D 2γ 2,X pr0, T s 2 , R dˆd q admit a smooth approximation. Then where f is the C 2γ function appearing in ( . ). In particular, note that one has a usual change of order of integration in the case where X is geometric: Proof. First note that all the double integrals are well-defined due to Lemma . . We will first prove the theorem for the case of geometric rough path and then will use decomposition ( . ) in order to show the general case. Let's call the left hand side of ( . ) L t and right hand side R t . The main idea is of approximation. Basically we want to show that:

R F T
Here X n and Y n P D 2γ 2,X n pr0, T s 2 , R dˆd q are as in the definition of smooth approximation. Since we can take X n to be geometric rough paths themselves then the middle equality in ( . ) is perfectly valid. This because in the smooth and geometric case rough integrals agree with the classical integrals for which the middle equality in ( . ) is certainly true.
It remains to establish the two other equalities. We will only show the third equality of ( . ). Once again since rough path X n is smooth and geometric then all the rough integrals with respect to X n are in fact the classical integral. We denote I n t " ş r 0 Y n r,s dX n s . From Lemma . both V and V n are controlled rough paths (respectively w.r.t. X and X n ) and for Gubinelli derivative of V we write 9 V r " Y r,r`ş r 0 Y 1 r,s dX s , and similarly 9 V n (In fact here you can see why does Y 1 and Y 2 also have to be rough paths). First write M " maxt̺ γ pXq, }pY, Y 1 q} X,2γ u. Because of convergence we can guarantee that eventually ̺ γ pX n q`}pY n , Y 1 n q} X n ,2γ ď 3M and so using stability of integration similar to Lemma . we get: Now first three terms clearly converges to 0 by approximation assumption. For the term | 9 V´9 V n | γ we can again use stability of integration and nice approximation assumption on Y to deduce that | 9 V´9 V n | γ Ñ 0 as n Ñ 8. We will show in more details on how to treat |R V´RV n | 2γ term. Therefore we have: The last two terms are bounded by Here |Ξ n v,u | 3γ " sup 0ďsătďT |Ξ n v,u pt,sq| |t´s| 3γ with Ξ n v,u pt, sq " Rpt, s, v, uqpX t,s´X n t,s q`pRpt, s, v, uq´Rpt, s, v, uq n qX n t,s δR 2,1 v,u pt, sqpX t,s´X n t,s q`pδR 2,1 v,u pt, sq´δR 2,1,n v,u pt, sqqX n t,s . We see that from approximation assumptions we indeed have For the remaining terms in equality for R V v,u´R V n v,u we add and subtract Y v,u X v,uÝ n v,u X n v,u . Then we use similar bounds for integrals to deduce: 2,n 0,0 |`}Y 2´Y 2,n } 8,γ`} R 2´R2,n } 8,2γ .
All terms converge to zero by assumption and so we are done proving the third equality in ( . ). Proving the first equality of ( . ) may seem to be more difficult since the integral inside is also t dependent. But in fact it is easy to check that it plays almost no role but requires a bit more computations similar to above. Thus we finish showing formula ( . ) for the geometric rough path. For the non geometric rough path the middle equality of ( . ) is no longer true because of the presence of the correction term f . In fact using the integral formula ( . ) and Lemma . , denoting by f n a smooth approximation of f we can show that: V n r¨d f n r .
Using that ş t 0 ş r 0 Y n r,s dX n s dX n r " ş t 0 V n r X n r and putting all the above formulas together we get that: Similarly Therefore using Y 1,2 " Y 2,1 we get: Letting n go to infinity we indeed get ( . ).

Theorem . (Rough Fubini Theorem).
Let γ P p1{3, 1{2s, X P C γ pr0, T s, R d q and Y P D 2γ 2,X pr0, T s 2 , R dˆd q admitting a smooth approximation. Then for rs, ts Ď r0, T s and ru, vs Ď r0, T s, one has the identity One can prove this theorem using the same argument of approximation and it is even easier to show than the Theorem . . Notice that when say in both integrals limits of integration are from 0 to t then Theorem . is a corollary of Theorem . . This is because the controlled rough path p ş t 0 Y r,m dX m , ş t 0 Y 1 r,m dX m q is a sum of two controlled R F T rough paths p ş r 0 Y r,m dX m , Y r,r`ş r 0 Y 1 r,m dX m q and p ş t r Y r,m dX m ,´Y r,r`ş t r Y 1 r,m dX m q for every r P r0, ts. Thus splitting the rough path p ş t 0 Y r,m dX r , ş t 0 Y 2 r,m dX r q similarly we get: A natural question to ask is whether a double integral of Y P D 2γ 2,X pr0, T s 2 , R dˆd q is itself an element of D 2γ 2,X pr0, T s 2 , Rq. We give the answer below but we do not study this question in much details since only Theorem . is needed for our purposes.
2,X pr0, T s 2 , Rq. Remark . . One can easily generalise the above results to functions in more time variables, giving rise to a generalised spaces D 2γ k,X pr0, T s k , R d k q for k P N. Another approach of defining the double integral is the approach of so called "Rough sheets" introduced in [CG ]. However, to best of our knowledge, no statement like Theorem . is known for Rough sheets.
It will also be useful to be able to rough integrals with usual Riemann integrals. Let Y : r0, T s 2 Ñ R d be a process such that for each fixed s P r0, T s, Y¨, s P D 2γ X pr0, T s, R d q is a controlled rough path and Y s,¨P Cpr0, T s, R d q is a continuous function. For such Y we say that it admits a smooth approximation if there exist sequences X n P C 8 pr0, T s, R d q, f n P C 8 pr0, T s, R dˆd q like in Definition . and Y n : r0, T s 2 Ñ R d such that for each fixed s P r0, T s, Y n ,s P D 2γ X n pr0, T s, R d q and Y n s,¨P Cpr0, T s, R d q such that where X n is a smooth function such that ̺ γ pX, X n q Ñ 0 as n Ñ 8. The following theorem can then be proved using the same method as Theorem . .

Theorem . .
Let γ P p1{3, 1{2s, X P C γ pr0, T s, R d q. Let Y : r0, T s 2 Ñ R d be such that for each fixed s P r0, T s, Y¨, s P D 2γ X pr0, T s, R d q and Y s,¨P Cpr0, T s, R d q. Assume that Y admits a smooth approximation as described above. Then we can perform the following exchange of the integrals:

W I ' RPDE
Note that no correction term with f from decomposition ( . ) arise in this case. This is because in the left hand side the rough integrand has a Gubinelli derivative Y 1 r,s (meaning in the first time variable) and the rough integrand in the right hand side has a Gubinelli derivative ş r 0 Y 1 r,s ds which will not create any correction terms when proving the analogue of the middle equality of ( . ).

Weak formulation and Itô's formula for RPDEs
In this section we are going to give an equivalent notion of solution for ( . ) -the weak solution. Recall that in Theorem . where we obtain solutions to the fixed point problem ( . ), we used the spaces D 2ε,2ε,ε S,X pr0, T s, H´2 γ q for 0 ă ε ă γ in order to obtain suitable bounds on the term }F pu t q´S t F pξq} H2ε´2γ . On the other hand, the right hand side of ( . ) makes sense as an element of H for any controlled rough path pu, u 1 q P D 2γ,2γ,0 S,X pr0, T s, H´2 γ q. This motivates us to give the following notions of solution: Definition . . Let γ P p1{3, 1{2s and X " pX, Xq P C γ pR`, R d q. Let ξ P H, F P C 2 2γ,0 pH, H d q, and N P Poly 0,n 0,´δ pHq for some n ě 1 and 1´δ ą γ. We say that pu, F puqq P D 2γ,2γ,0 S,X pr0, T s, H´2 γ q is a mild solution of the equation if, for each 0 ď t ď T , the following identity holds: We say that pu, F puqq P D 2γ,2γ,0 S,X pr0, T s, H´2 γ q is a weak solution if for every h P H 1 and 0 ď t ď T the following integral formula holds: xu t , hy " xu 0 , hy`ż t 0 xu s , Lhyds`ż t 0 xN pu s q, hyds`ż t 0 xF pu s q, hydX s . ( . ) Note that since N pu s q P H´δ and δ ă 2{3, xN pu s q, hy is well-defined. Moreover, since 2γ ă 1 and therefore H 1 Ď H 2γ , Proposition . guarantees that xF pu s q, hy is a controlled rough path in the classical sense and the integral ş t 0 xF pu s q, hydX s is welldefined. We are going to prove that these two notions of solution are in fact equivalent. To prepare this proof, we have the following preliminary result: Lemma . . Let X P C γ pr0, T s, R d q for γ P p1{3, 1{2s. Then for every h P H and pY, Y 1 q P D 2γ,2γ,0 Proof. First note that by Remark . and since by Proposition . D 2γ,2γ,0 S,X " D 2γ,2γ,0 X , we can find a smooth approximation of pY, Y 1 q, meaning that there exists a sequence of X n " pX n , X n q P C γ with X n smooth such that ̺ γ pX, X n q Ñ 0 as n Ñ 8 and a sequence pY n , Y 1 n q P D 2γ,2γ,0 S,X n pr0, T s, H d 2γ q such that d 2γ,2γ,0 pY, Y n q Ñ 0 as n Ñ 8.

W I ' RPDE
By Proposition . , W n t,r " ş t r xS s´r Y n r , hyds is a controlled rough path with respect to X n and W t,r " ş t r xS s´r Y r , hyds is a controlled rough path with respect to X. Therefore the following integrals can be defined in the rough path sense: Z n t " ş t 0 W n t,r dX n r , Z t " ş t 0 W t,r dX r . (The fact that W also depends on t does not cause any difficulties in defining the integral). Similar arguments as in the proof of Theorem . allow us to deduce that }Z´Z n } 8,r0,T s À pd 2γ,2γ,0 pY, Y n q`̺ γ pX, X n qq}h} Ñ 0 as n Ñ 8 , thus Z n Ñ Z uniformly in time. Moreover we know from the stability of integration Lemma . that for V n s " ş s 0 S s´r Y n r dX n r and V s " ş s 0 S s´r Y r dX r we have: }V n´V } 8,H À d 2γ,2γ,0 pY, Y n q`̺ γ pX, X n q Ñ 0 as n Ñ 8 .
It is easy to see that smoothness of X n implies ş t 0 xV n s , hyds " Z n t and thus:ˇˇż Proof. Without loss of generality we can assume in both cases that ξ " 0 by replacing pu, F puqq by pu`S¨ξ, F puqq (usingδS¨ξ " 0). Mild ñ Weak. Assume also for simplicity that N " 0 since dealing with the drift term term is easier than with the diffusion term F . Now let pu, F puqq P D 2γ,2γ,0 S,X pr0, T s, H´2 γ q satisfy for 0 ď t ď T u t " ż t 0 S t´s F pu s qdX s .
Let h P H 1 be arbitrary. Then taking the inner product with Lh and integrating from 0 to t gives where we used Lemma . in the second equality together with Lh P H. To conclude, it suffices to note that by Proposition . xF pu r q, S t´r hy " xS t´r F pu r q, hy is itself a rough path and therefore ż t 0 xF pu r q, S t´r hydX r " x ż t 0 S t´r F pu r qdX r , hy " xu t , hy .

W I ' RPDE
Weak ñ Mild. The proof is almost identical to the standard proof for SPDEs and can be found either in [Hai ] or [DPZ ].
The next lemma is a slight generalisation of Theorem . , but it has exactly the same proof, so we omit it.
Note that h P H 1´α`σ guarantees that xv s , Lhy is well-defined because of v s P H α´σ and Lh P H´α`σ.
A particularly important case is the choice v t " Apu t q for some regular function A P C 2 α´2γ,´σ pHq. By Lemma . , for every pu, u 1 q P D 2γ,2γ,0 S,X pr0, T s, H α´2γ q, we then have pApuqq, DApuqu 1 q P D 2γ,2γ,0 S,X pr0, T s, H α´2γ´σ q. The question we want to ask is whether Apu t q satisfies some mild formula like ( . )? Before we answer this question we recall the definition of the bracket of a rough path: Definition . . Let V be a Banach space and X P C γ pr0, T s, V q, then its bracket is given by rXs t " X t,0 b X t,0´2 SympX t,0 q.
From Chen's relation ( . ) it follows that δrXs t,s " X t,s b X t,s´2 SympX t,s q , and therefore rXs P C 2γ pr0, T s, V q. In particular rXs " 0 if and only if X is a geometric rough path. Moreover the above implies that rXs t "´2Sympf t q where f P C 2γ is as in the decomposition ( . ). For example rB Itô s t " t and rB Strat s t " 0 almost surely. Now for a moment assume that rXs " 0 so that there is no "Itô correction" and we can just apply the chain rule. Assume that A is Fréchet differentiable and u formally satisfies an equation du t " Lu t dt`N pu t qdt`F pu t qdX t . Then heuristically we have: dpApu t qq " DApu t qdu t " DApu t qLu t dt`DApu t qN pu t qdt`DApu t qF pu t qdX t " LApu t qdt``DApu t qN pu t q`rL, Aspu t q˘dt`DApu t qF pu t qdX t .
Since L is linear, we have DLpuq " L for each u P H α , therefore rL, Aspu t q " DApu t qLu t´L Apu t q. WritingÑ puq " DApuqN puq`rL, Aspuq andF puq " DApuqF puq then on a formal level Apu t q solves dpApu t qq " LpApu t qqdt`Ñ pu t qdt`F pu t qdX t .

W I ' RPDE
This suggests that Apu t q satisfies the identity Before showing this result for the mild formulation rigorously we state a weak version of it: Theorem . (Weak Itô formula). Let γ P p1{3, 1{2s, X P C γ , σ ě 0, and α P R.
Setting v 1 t " F pu t q and v 2 t " DF pu t qF pu t q, we note that pG, G 1 q " pDApvqv 1 , D 2 Apvqpv 1 , v 1 q`DApvqv 2 q is itself a controlled rough path in D 2γ X pr0, T s, H α´2γ´ν q. As in [FH , Remark . ], one can define V t,s " ş t s pv r´vs q b dv s P C 2γ 2 pH α´2γ b H α´2γ q, yielding a rough path v " pv, Vq P C γ pH α´2γ q. From now on we are going to omit writing b and always understand say φψ for two elements of some Hilbert space φ and ψ as their tensor product φ b ψ. By Itô's formula for rough paths [FH , Prop. . ] we have that: δApvq t,0 " Apv t q´Apv 0 q " lim With convergence in H α´2γ´ν . Now one can show that δrvs r,m " v 1 m v 1 m δrXs r,mò p|r´m|q and using that V r,m " v 1 m v 1 m X r,m`o p|r´m|q we can take an inner product of ( . ) with h P H 1´α`ν on both sides we get: xδApvq t,0 , hy " lim

W I ' RPDE
The second term in this expression converges to 1 2 ş t 0 xD 2 Apv s qpv 1 s , v 1 s q, hydrXs s , interpreted as a Young integral. Since v 1 s " F pu s q, this gives the very last term in ( . ). To deal with the first term in ( . ), note that for a fixed value of m, one has xDApv m qδv r,m , hy " xδv r,m , DA˚pv m qhy with DA˚pv m qh P H 1´α , so that we can apply ( . )  We conclude that one has I r,m " xDApv m qLv m`D Apv m qN pu m q, hypr´mq xG m X r,m`G 1 m X r,m , hy`op|r´m|q.
Since pG, G 1 q is a controlled rough path, we then obtain We conclude by recalling that ş t 0 G s dX s " ş t 0 DApv s qv 1 s dX s " ş t 0 DApv s qF pu s qdX s .
Proof. By Lemma . , equation ( . ) holds for pv, F puqq, so that ( . ) holds for every h P H 1´α`σ`ν by Theorem . . We now make use of the fact that xDApv s qLv s`D Apv s qN pu s q, hy " xLApv s q, hy`xDApv s qN pu s q`rL, Aspv s q, hy B RPDE " xApv s q, Lhy`xDApv s qN pu s q`rL, Aspv s q, hy, where xrL, Aspv s q, hy makes sense since rL, Aspv s q P H α´σ´ν´1 and h P H 1´α`σ`ν . Thus we get the following weak equation: xApv t q, hy " xApv 0 q, hy`ż t 0 xApv s q, Lhyds`ż t 0 xDApv s qN pu s q`rL, Aspv s q, hyds ż t 0 xDApv s qF pu s q, hydX s`1 2 ż t 0 xD 2 Apv s qpF pu s q, F pu s qq, hydrXs s , which itself implies the mild formula ( . ) by Lemma . and the fact that the last integral is well-defined as a Young integral.

Backwards RPDEs
We will briefly describe the method of solving rough backwards PDEs of the form: For short we call them backwards RPDEs. We will quickly describe the theory of backwards controlled rough paths according to the semigroup. In many instances, the proofs of the results are virtually identical to the corresponding ones for forward controlled rough paths, so we do not give them. We introduce an increment operatoř δ : for a semigroup S acting on a Banach space V . (We will actually assume that S consists of selfadjoint operators on some Hilbert space H.) With this, we define a Hölder like spaceČ γ " tf P C 1 : |δf | γ,V ă 8u , and we endow it with a seminorm~f~γ ,V " |δf | γ,V and a norm }f }Čγ "~f~γ ,V} f T } V . (We could have replaced }f T } V by }f } 8,V , which yields an equivalent norm.) Definition . . Let X P C γ pr0, T s, R d q for some γ P p1{3, 1{2s and let m P N. We say that pY, Y 1 q PČ γ pr0, T s, H m α qˆČ γ pr0, T s, H mˆd α q is backwards controlled by X according to the semigroup pS t q tě0 if the remainder term defined through This defines a space of controlled rough paths (according to the semigroup) We endow this space with a semi-norm (omitting d and m for notational convenience) It is easy to see that the space D 2γ S,X,Ð pr0, T s, H m α q is a Banach space with norm:

B RPDE
Here the endpoint Y T plays the same role as the starting point for forward controlled rough paths. This is justified by the inequality }Y } 8 À T }Y T }`~Y~γ. This also corresponds to the fact that for backwards RPDEs we don't know the initial condition but rather the terminal condition.
Similarly as for forward controlled rough paths for β P R and η P r0, 1s define a space We introduce a norm on this space to be: Here we also make an abuse of notation by writingČ 0 " L 8 for η " 0. Similarly to Lemma . , composition with regular functions maps D 2γ,2γ,η S,X,Ð pr0, T s, H α q to D 2γ,2γ,0 S,X,Ð pr0, T s, H α q for every η P r0, 1s.
For pY, Y 1 q P D 2γ S,X,Ð pr0, T s, H d α q an integral ş T t S r´t Y r dX r can be defined and Moreover, results analogous to Theorem . , Lemma . , and Theorem . are true and their proofs are almost the same. The main difference is that the role of the initial condition Y 0 is now played by the terminal condition Y T . We can now state a theorem regarding solutions to backwards equations of the type arising in ( . ).

Theorem . (Nonlinear backwards RPDEs).
Let γ P p1{3, 1{2s and X " pX, Xq P C γ pR`, R d q. Then, given ξ P H, F P C 3 2γ,0 pH, H d q, and N P Poly 0,n 0,´δ pHq for some n ě 1 and 1´δ ą γ, there exists τ ě 0 and a unique element pv, v 1 q P D 2γ,2γ,γ S,X,Ð ppτ, T s, H´2 γ q such that v 1 " F pvq and v t " S T´t ξ`ż T t S r´t N pv r qdr`ż T t S r´t F pv r qdX r , v T " ξ P H.
We call the pair pv, F pvqq the mild local solution to the backwards RPDE dv t "´Lv t dt´N pv t qdt´F pv t qdX t and v T " ξ P H.
For a weak solution approach to both forward and backward rough PDEs we refer the reader to [DFS ].
One can show that all the continuity results of Section . are true for backwards RPDEs. The same is true for a smoothing result analogous to Proposition . , except that smoothing now takes place away from the terminal point v T " ξ. One can show that solutions to backwards RPDEs coincide with solutions to backwards SPDEs in the case of Brownian motion. From now on, we will assume that we are in the setting of Theorem . with choices of L, N , F and X such that one can choose τ " 0, so that solutions exist (and are unique) on the whole of r0, T s. The following proposition establishes a connection between the forward and backward controlled rough paths.
Proof. The proof is a straightforward computation where we use the fact that S t is a selfadjoint operator on H for any time t ě 0. By the definition of controlled rough path xV t , Z t y´xV s , Z s y " xV t , Z t y´xS t´s V s , Z t y`xV s , S t´s Z t y´xV s , Z s y " pxS t´s V 1 s , Z t y`xV s , S t´s Z 1 t yqX t,s`x R V t,s , Z t y`xV s , R Z t,s y.

Now
xS t´s V 1 s , Z t y`xV s , S t´s Z 1 t y " Y 1 s`x V 1 s , S t´s Z t´Zs y`xV s , S t´s Z 1 t´Z 1 s y.
We can therefore write The bound ( . ) is then an easy consequence of decomposition above. The requirement on exponents α and β is necessary since we want to bound terms like: Here we need α`β`2γ ě 0 so that we can use the Cauchy-Schwarz inequality.
We just showed that the inner product of a forward controlled rough path with a backward controlled rough path is a controlled rough path in the usual sense. Assuming that these controlled rough paths solve respectively some RPDE and backwards RPDE in the mild sense, we can ask ourselves whether their inner product also satisfies an integral equation. It turns out that this is true and this inner product in fact solves an RDE: Proposition . . Let 1{3 ă ε ă γ ă 1{2 and X P C γ pr0, T s, R d q. Let δ ď 1 and simultaneously α`β`4γ´δ ě 0 and α`β`2γ ě 0. Let V P D 2ε,2γ,0 S,X pr0, T s, H α q and Z P D 2ε,2γ,0 S,X,Ð pr0, T s, H β q be such that they satisfy these mild forward and backward equations on r0, T s: Then Y t :" xV t , Z t y P D 2ε X pr0, T s, Rq is a controlled rough path that satisfies the following integral formula: where the function f P C 2γ pr0, T s, R dˆd q is the one appearing in the decomposition ( . ) of the rough path X. In particular, if X is geometric andÑ ,F are the adjoints of N and F , then Y t is constant in time: Y t " xV 0 , Z 0 y " xY T , Z T y for every t P r0, T s.

Proof.
Note that the assumptions on α, β and δ are necessary for all the integrals in ( . ) to make sense. (If δ " 1 we only need the assumption α`β`4γ´δ ě 0). We will assume that N "Ñ " 0 since the drift term dt is even simpler to treat than the dX t term. (Note though that Theorem . needs to be used at some point to swap the order of integration in integrals of the type ş t 0 ş t s xN r , S r´sFs y dr dX s ). Using the mild equations for V t and Z t we get: We can move the inner products inside the integration by examining the proof of the Sewing Lemma, Theorem . , and ideas similar to Proposition . . Since S is selfadjoint, we get: We will show that R t " 2 ş t 0 xF s ,F s y¨df s for every t, which then implies the result. One has the identity Setting W r,s " xF s , S r´sFr y, we would like to show that one can apply our version of Fubini's theorem, Theorem . . We are almost in the situation of Theorem . : the only difference is that W r,s is defined only for r ě s because of the presence of the semigroup S r´s . But if one examines the proof of Theorem . , one can see that we can always require that r ě s in our computations. Here we have W 1 r,s " xF 1 s , S r´sFr y; W 2 r,s " xS r´s F s ,F 1 r y; W 1,2 r,s " xS r´s F 1 s ,F 1 r y.
The remainders R 1 , R 2 , R 1,2 , R 2,1 are also easy to determine. Since by Remark . both F andF admit a smooth approximation then so does W r,s in the sense of Definition . . Thus we can indeed swap the integrals for W r,s like in Theorem . , deducing: We conclude that R t " 2 ş t 0 xF s ,F s y¨df s and hence we are done.

. Adjoint of the Jacobian
From now on for simplicity we denote the Jacobian of the solution to the ( . ) by J t,s , omitting the reference to the noise X. In the later results we would like to use the adjoint of the Jacobian of the solution Jt ,s . For instance, this appears in the expression for the Malliavin matrix xM t ϕ, ϕy " ş t 0 xF pu s q, Jt ,s ϕyds. It would then be useful to know that Jt ,s also solves an RPDE. Unfortunately, having a mild formulation for J t,s is not enough to deduce a mild formulation for Jt ,s . Therefore we go the other way around: we 'guess' an equation for Jt ,s and then show that the solution to this equation is indeed the adjoint of the Jacobian. In fact it will be more convenient to work with the backwards equation for the adjoint. This is because Proposition . then gives us an explicit expression for xJ r,s ϕ, Jt ,r ψy for any ψ, ϕ P H. A natural guess is to take a backwards analogue of ( . ) where we formally take adjoints of the linear maps DN and DF , so that our ansatz for Jt ,s is: The next proposition shows that this guess is indeed correct.
Proposition . . Let X P C γ g pr0, T s, R dˆd q be a geometric rough path. Let pu, F puqq P D 2γ X pr0, T s, Hq be the solution to ( . ) with F and N as in Proposition . . For every t P r0, T s and every ϕ P H, let pK t,¨, K 1 t,¨q :" pK t,¨, DF˚pu¨qK t,¨q P D 2γ,2γ,γ S,X,Ð pr0, ts, H´2 γ q be the solution to the backwards equation K t,s ϕ " S t´s ϕ`ż t s S r´s DN˚pu r qK t,r ϕdr`ż t s S r´s DF˚pu r qK t,r ϕdX r . ( . ) Then K is the adjoint of the Jacobian: K t,s " Jt ,s for all 0 ď s ď t ď T . Proof. We want to show that xJ t,s ϕ, ψy " xϕ, K t,s ψy for all ϕ, ψ P H. Set Y r " xJ r,s ϕ, K t,r ψy and note that thanks to the smoothing property of the solutions, Proposition . , the regularity assumptions of Proposition . are satisfied for pY r , Y 1 r q P D 2γ X prs`ε, t´εs, Rq for all ε ą 0. Moreover, since X is geometric, Y t satisfies the equation Since the terms inside the integrals cancel each other, we have Y t´ε " Y s`ε , i.e. xJ t´ε,s ϕ, K t,t´ε ψy " xJ s`ε,s ϕ, K t,s`ε ψy.
But from the mild representation of K t,r and J r,s , we see that both of these lie in the space Cprs, ts, Hq as functions of the r variable. We can therefore take the limit of the above expression as ε goes to zero to obtain: xJ t,s ϕ, K t,t ψy " xJ s,s ϕ, K t,s ψy.
Recalling that J s,s ϕ " ϕ and K t,t ψ " ψ, we get the desired result. S M Proposition . . Let X P C γ g pr0, T s, R d q and γ P p1{3, 1{2s. Let pu, F puqq P D 2γ X pr0, T s, Hq be the solution to ( . ) with F and N as in Proposition . . Let K t,s be the adjoint of the Jacobian. Let ν ě 0 and a function A P C 2 0,´ν pHq. Fix 0 ď t ď T , ϕ P H and set Z ϕ A prq :" xApu r q, K t,r ϕy. Then Z ϕ A P D 2γ X prs, ts, Rq for every 0 ă s ď t and solves the RDE dZ ϕ A prq " Z ϕ rL`N,As prqdr`Z ϕ rF,As prqdX r , which in its integral form reads for all r P rs, ts: Proof. First we use the mild Itô formula Theorem . to determine the mild equation for Apu s q. Note that since X is geometric, its bracket rXs vanishes. We then use the mild formulation ( . ) for K t,s and the fact that it is arbitrarily smooth on rs, tq together with mild equation for u r and the fact that it is arbitrarily smooth on p0, ts to ensure that we can apply Proposition . to derive the equation for Z ϕ A prq " xApu r q, K t,r y. It is easy then to verify that this is indeed Equation . .

Spectral properties of the Malliavin matrix
For this section we consider a special case when the multiplicative noise term is given by where F i P C 8 2γ,0 pHq are smooth functions. We also assume N is smooth and belongs to Poly 8,n 0,´δ and consider the collections of Lie brackets A k defined recursively by: A 0 " tF i : 1 ď i ď du; A k`1 " A k YtrL`N, As, rF i , As : A P A k , 1 ď i ď du.
Note that, at worst, elements of A k decrease the spatial regularity by k i.e. send H k to H.
Our aim is to show that, under a version of Hörmander's condition appropriate for this context, one obtains a bound on the Malliavin matrix of the kind Ppinf ϕ xM T ϕ, ϕy ď εq À T,p ε p for every p ě 1 (we will specify precisely over which class of ϕ we take the infimum later). The proof is in the same spirit as the proof of Hörmander's theorem for SDEs using Malliavin calculus techniques, see for instance [Mal , Hai ]. It essentially goes by contradiction: assuming that xM T ϕ, ϕy is small, ( . ) then implies that xJ T,s F pu s q, ϕy is small. In the SDE case the solution to the equation with good enough vector fields generates a smoothly invertible flow, so it is possible to factor the Malliavin matrix as Then the process s Þ Ñ xJ´1 s,0 F pu s q, ϕy is a semimartingale and one can use Norris's lemma [Nor ] to deduce by induction over k that s Þ Ñ xJ´1 s,0 Apu s q, ϕy is small for every vector field A P A k . Hörmander's condition then guarantees that the span of the A k at every point is dense in H, which contradicts the fact that all the xJ´1 s,0 Apu s q, ϕy are small by considering s " 0.

S M
The problem with this argument is that solutions to parabolic SPDEs do not produce a smoothly invertible flow, so that the Jacobian J s,t is not invertible. In [HM ] where the authors deal with the case of additive noise and polynomial nonlinearities, they use a version of Norris's lemma for Wiener polynomials instead of semimartingales. In our setting, we consider rough integration instead of Itô integration, which allows us to use a version of Norris's lemma for rough paths. Before stating it, we recall the notion of Hölder roughness from [HP ]: Definition . . Let θ P p0, 1q, a path X : r0, T s Ñ R d is said to be θ-Hölder rough if there exists a constant L θ pXq such that for all s P r0, T s, all ε P p0, T {2s and every z P R d with |z| " 1, there exists a t P r0, T s such that |t´s| ď ε and |pz, X t,s q| ą L θ pXqε p .
We denote the largest such L θ pXq the modulus of θ-Hölder roughness of X.
Here px, yq denotes the scalar product on R d . In [HP ] it was proved that if X is a fractional Brownian motion with Hurst parameter H ď 1{2, its sample paths are almost surely θ-Hölder rough for every θ ą H. Moreover, there exist constants M and c independent of X such that for every ε P p0, 1q, so that in particular ErL´p θ pXqs ă 8 for every p ě 0.
With this at hand we are ready to state one more result from [HP ], namely the aforementioned version of Norris's lemma for rough paths.
Theorem . . Let γ P p1{3, 1{2s and pX, Xq P C γ pr0, T s, R d q be θ-Hölder rough for θ ă 2γ. Let V P C γ pr0, T s, Rq and pY, Y 1 q P D 2γ X pr0, T s, R d q and set Then there are constants q ą 0 and r ą 0 such that, setting we have the bound }Y } 8`} V } 8 À T R q }Z} r 8 on r0, T s.
We will now work with a solution pu, u 1 q to ( . ) starting from u 0 P H driven by the path pX, Xq P C γ g pR`, R d q and vector fields N, F i as described in the beginning of this section. K¨,¨denotes the adjoint of Jacobian as in Section . Fix T and a smaller time 0 ă s ă T , let 1{3 ă η ă γ ă 1{2 be such that η is close to γ. From now on fix the quantity: R s pu 0 q " 1`L θ pXq´1`̺ γ pXq`}pu, u 1 q} D 2η,2γ,0 S,X prs,T s,Hαq }pK T,¨, K 1 T,¨q } D 2η,2γ,0 S,X,Ð prs,T s,L´2 γ q , ( . ) for α ě 0 big enough to be determined later. If any of the quantities above explodes on the interval rs, T s we simply write R s pu 0 q " 8. As in Proposition . define for ϕ P H and a vector field A, a function Z ϕ A prq " xApu r q, K T,r ϕy. From Proposition . and Lemma . it follows inductively that for S M every k P N 0 there exists a constant C k depending on T and L, N, F i such that for all A P A k we have: The above holds true if in the definition of R s we take α big enough (depending on k) so that the assumptions on spatial regularities of Proposition . would be satisfied.

Remark . .
Note that we only impose high spatial regularity on the solution u r and not on K T,s . This will give us the advantage of being able to use the fact that K T,T is the identity.
The following two results are almost exact analogues of the finite-dimensional statements from [HP ]. Proposition . allows us to carry out the same techniques.
Lemma . . Let T ą 0 then for all 0 ă s ă T , there exist q, r ą 0 and M independent of X, ϕ, u 0 such that for all A P A 0 the following bound holds: ) for all ϕ P H such that }ϕ} " 1 and all initial conditions u 0 P H.

Proof. Note that we have
Consider an interpolation inequality (see [HP ] Lemma A. ): Since the final time is fixed, the L 2 norm is controlled by the γ-Hölder norm, so }Z ϕ Fi } 2γ`1 8,rs,T s À s }Z ϕ Fi } 2γ L 2 rs,T s |Z ϕ Fi | C γ rs,T s .
The first term is clearly controlled by xM T ϕ, ϕy 2γ since rs, T s Ă r0, T s and the second term is controlled by C 0 R s pu 0 q 2 by ( . ). Now we show that the same holds for any vector field in A k .
Lemma . . Let T ą 0 and pX, Xq P C γ g pr0, T s, R d q be θ-Hölder rough for θ ă 2γ. Then for all 0 ă s ă T and every k P N 0 there exist q k , r k ą 0 and M k independent of X, ϕ, u 0 such that for all A P A k the following bound holds: for all ϕ P H such that }ϕ} " 1 and all initial conditions u 0 P H.

Proof.
Define first for A P A k a quantity R A " 1`L θ pXq´1`̺ γ pXq`|pZ ϕ rF,As , pZ ϕ rF,As q 1 q| X,2γ`| Z ϕ rL`N,As | C γ , with all the norms taken on the interval rs, T s. Note that from ( . ) we have R A ď C k R 2 s pu 0 q uniformly over all }ϕ} " 1. Assume now that the statement is true for k.

S M
Let A P A k , denote F 0 " L`N then by Proposition . we have the representation for all s ď r ď T : Therefore we can apply the "rough Norris lemma", Theorem . , to deduce: Since all B P A k`1 are of the form rF i , As for i " 0, . . . , d and A P A k we conclude by induction.

Remark . .
Note that both of the above lemmas are purely deterministic. Moreover we did not make any assumption on the solution or its Jacobian. In fact if the solution explodes before time T or if it does not have a Jacobian on the interval rs, T s one should simply read the inequalities ( . ) and ( . ) as trivial statements "8 ď 8".
We will now present the precise assumptions on the noise and solution which will enable us to prove our Hörmander's theorem. We start with an assumption on the driving noise which guarantees that it gives rise to a well-behaved rough path, but is also sufficiently irregular to kick the solution around in a very non-degenerate way.
Assumption A. . For some γ P p 1 3 , 1 2 q, the random rough path X " pX, Xq P C γ g pr0, T s, R d q is the canonical lift of a d-dimensional, continuous Gaussian process X with independent components defined on some underlying probability space pΩ, F , Pq. We also assume that there exist M ă 8 and p P r1, 1{2γq such that for i P t1, . . . , du and rs, ts Ď r0, T s, the covariances of X i satisfy }R X i } p,rs,ts 2 ď M |t´s| 1{p .
We also assume that X is almost surely θ-Hölder rough for some θ ă 2γ and moreover that all inverse moments of its modulus of θ-Hölder roughness are bounded, i.e. ErL´q θ pXqs ă 8 for all q ě 1.
With the driving noise X at hand, we assume the global existence of the solution: Assumption A. . Let X be as in Assumption A. and let tF i u d 1 Ă C 8 2γ,0 pHq, N P Poly 8,n 0,´δ pHq for δ ă 1´γ. We assume that for every initial condition u 0 , ( . ) has a global solution pu, F puqq P D 2γ X pR`, Hq for almost every realisation of X. We also assume that the Jacobian J t,s and its adjoint K t,s exist for all times and satisfy the corresponding mild equations ( . ) and ( . ) respectively.
In the parabolic case, we cannot expect to have a bound on inf }ϕ}"1 xM T ϕ, ϕy since this would imply the invertibility of the Malliavin matrix, contradicting the fact that M T is a compact operator. Instead, we consider an orthogonal projection Π : H Ñ H with finite-dimensional range and, for a P p0, 1q, we define S a Ă H to be S a " tϕ P H : }ϕ} " 1, }Πϕ} ě au.
For k P N 0 define the positive symmetric quadratic form-valued function Q k such that for all u P H 8 xϕ, Q k puqϕy " ÿ APA k xϕ, Apuqy 2 .

S M
With this notation we assume that the following non-degeneracy condition holds on the Lie brackets in A k : Assumption A. . Assume that tF i u d 1 Ă C 8 2γ,0 pHq, N P Poly 8,n 0,´δ pHq, δ ă 1´γ. Moreover assume that for some orthogonal projection Π : H Ñ H and for every 1 ą a ą 0, there exists k P N 0 as well as a continuous function Λ a : H Ñ p0, 8q such that inf ϕPSa xϕ, Q k puqϕy ě Λ a puq, for every u P H 8 .
Finally, we assume that we have good enough control on the solution to be able to "fight" the loss of control generated by regions where the function Λ a from Assumption A. is small. A. , A. , A. hold and that there exist two functions Φ 1 , Φ 2 : Hˆr0, 8q Ñ r0, 8q such that the following two growth assumptions are true:

Assumption A. . We assume that Assumptions
( ) For all p ě 1 and all a P p0, 1q there exist K 1 such that for the solution u to ( . ), the inverse moment bound ErΛ´p a pu T qs ď K 1 Φ p 1 pu 0 , T q, holds for every initial condition u 0 P H.
The requirement for α is coming from the fact that for all A P A k the inner product xApu s q, K T,s ϕy should satisfy the assumptions of Proposition . on spatial regularities and the fact that the assumption of Lemma . for N should be satisfied (N has to be a smooth function on the level of rough path regularity, which is α in this case).
Note that among all the assumptions we do not a priori assume that the solution is Malliavin differentiable, but this does follow from assumptions A. and A. , combined with Theorem . . Moreover, Definition . of M T does not coincide with the definition of the Malliavin matrix in general (see [Nua ]), but it always makes sense whenever Jacobian is well-defined. In the particular case where X is a Brownian motion, we will see in the proof of Theorem . below that two definitions agree and our version of Hörmander's theorem provides a statement for the usual Malliavin matrix. With all these assumptions at hand we are ready to present the main result of this article. Theorem . . Let T ą 0 and let the noise X P C γ g satisfy Assumption A. . Let 0 ď δ ă 1´γ and assume that N and F i satisfy Assumption A. for some orthogonal projection Π : H Ñ H, a P p0, 1q, k P N 0 and continuous function Λ a : H Ñ p0, 8q. Assume that pu, F puqq solving ( . ) satisfies Assumptions A. and A. . Then there exists a function Φ T : H Ñ r0, 8q such that for every p ě 1 there exist a constant C p such that the operator M T defined in ( . )

satisfies the bound
Pp inf ϕPSa xM T ϕ, ϕy ď εq ď C p Φ p T pu 0 qε p , for every initial condition u 0 . Here C p is independent of the initial condition.

S M
Proof. Fix ϕ P S a , an initial condition u 0 P H, and let A k and Q k be as in Assumption A. . Since K T,T ϕ " ϕ and, by Proposition . , u T P H 8 almost surely, we have: Λ a pu T q ď xϕ, Q k pu T qϕy À max where R T {2 pu 0 q is defined by ( . ) with α " pk´2γ`δq_ 0. This shows the existence of some q, r, M ą 0 independent of noise and initial condition such that for all ϕ P S a xM T ϕ, ϕy ě M Λ r a pu T qR´q T {2 pu 0 q.
Here we used Cauchy-Schwarz in the third inequality and, in the last inequality, we used A. . Finally from A. the expectation ErL´2 pq θ pXq`̺ 2pq γ pXqs is always bounded by some constant C p and thus taking for instance gives the result.
We now give an extra condition on the solution u which will guarantee the smoothness of the densities of finite-dimensional projections: Assumption A. . In addition to previous assumptions assume that for all T ą 0 there exists a function Ψ T : H Ñ r0, 8q such that for all p ě 1 there exist K 3 with: for every initial condition u 0 and function Φ from the Theorem . .

Theorem . .
Assume that the rough path X " pB, B Strat q. Let pu, u 1 q be a global solution to ( . ) like in Theorem . , and assume that it also satisfies A. . Assume also that the image of the orthogonal projection Π (from assumption A. ) is finitedimensional. Then for all t ą 0 the law of Πu t has a smooth density with respect to Lebesgue measure on ΠpHq.
Proof. Since ( . ) is driven by the Stratonovich lift of Brownian motion to the space of rough paths, it follows from Proposition . that u coincides almost surely with the solution to the corresponding Itô SPDE. It follows from the growth assumption A. that we can derive an SPDE for the Malliavin derivative of any order and the Jacobian of any order. Moreover, using Duhamel's formula similar to ( . ) for higher order Malliavin derivatives and moment assumptions from A. one concludes that for every t ą 0, u t belongs to the space D 8 of Malliavin smooth random variables whose Malliavin derivatives of all orders have moments of all orders. (For more details in the additive case see [HM , Thm . ].) Since u t is Malliavin smooth and Π is a bounded linear map (hence smooth), we deduce that Πu t is also Malliavin smooth. By [Nua , Chap. ] it remains to show that the Malliavin matrix of Πu t denoted by M Π t is almost surely invertible and has moments of all orders. Just for notational convenience we will consider Πu 2t instead of Πu t . First we can view the element u 2t as an element of the probability space with Gaussian structure induced by the increments of B over the interval rt, 2ts and, as in [Nua , Chap. ], we view increments of B over r0, ts as irrelevant randomness. This shows that almost surely M Π 2t " ΠM t pu 0 qΠ`ΠM t,2t pu t qΠ ě ΠM t,2t pu t qΠ , where M t,2t pu t q is defined like in ( . ) but over the interval rt, 2ts, and we treat u t as an "initial condition" at time t. Recall that F s is the natural filtration of the underlying Brownian motion, denote K " ΠpHq then we have: Pp inf ϕPK; }ϕ}"1 xΠM t,2t pu t qΠϕ, ϕy ď ε|F t q " Pp inf ϕPK; }ϕ}"1 xM t,2t pu t qϕ, ϕy ď ε|F t q ď Pp inf ϕPSa xΠM t,2t pu t qΠϕ, ϕy ď ε|F t q ď C p Φ p t pu t qε p .
Moreover in the last inequality we have used the Markov property of the solution.
This guarantees the invertibility of Malliavin matrix M Π 2t on K and that pM Π 2t q´1 has moments of all orders, thus finishing the proof.

Examples
As mentioned before, we will restrict ourselves to the Brownian rough path in all examples since Assumption A. is a priori not known to hold for more general Gaussian rough paths. We will focus on equations driven by Brownian motion that have global solutions as well as a Jacobian, which will imply Assumption A. from Section . . We will then show examples of noises for which Hörmander's condition, Assumption A. is satisfied. The moment bounds for the rough path norms of solution and Jacobian for Assumption A. part ( ) might not be easy to obtain in general and require a closer look as a separate problem on its own. We decide to postpone the study of such moments but refer the reader to [FR ] where this question was answered for the rough SDE case.
We want to point out that the present work is indeed a generalisation of the additive case from [HM ] since Assumption A. is a slight modification of Assumption C. from that article. Consider an equation which has both an additive and a multiplicative noise: for g i P H 8 . Note that as a Fréchet derivative Dg i " 0 thus the Lie brackets rg i , g j s " 0 and there is no contribution from Lie brackets of this additive part. Only an interplay of rL`N, g i s, rL`N, F i s, rF j , g i s, rF j , F i s and of higher order Lie brackets contributes to Assumption A. . In particular, if ( . ) satisfies Assumption A. for F i " 0, then it also satisfies it for F i ‰ 0.
We now give a simple criteria for when the Assumption A. is satisfied and moreover the function Λ a can be taken constant. A " ď kě0 tA P A k : @u P H k , Apuq " Ap0q P Hu.
If the linear span of A is dense in H, then for every finite rank orthogonal projection Π : H Ñ H and every a P p0, 1q Hörmander's condition A. is satisfied for some k. Moreover, the function Λ a can be chosen as a constant depending on Π and a. As a consequence, part p1q of the condition A. is trivially satisfied too.
A proof of this statement can be found in [HM , Lem. . ]. This criterion is the one that we are going to use in our next examples. ( . ) is driven by Brownian motion and solutions u t satisfy Assumption A. , then the above proposition and Theorem . guarantee that Πu t has a smooth density with respect to Lebesgue measure for every surjective linear map Π : H Ñ R n .

. Stochastic Navier-Stokes equation in -d
The Navier-Stokes equation describes the time evolution of incompressible fluid and is given by B t u t pxq`pu t pxq¨∇qu t pxq " ∆u t pxq´∇p t pxq`ξpt, x, u t pxqq, ∇¨u t " 0.
For u t pxq P R 2 is a velocity field, ppt, xq is a pressure, and ξ is a noise term describing an external force acting on the fluid. We decide to work with the equation with spatial variable lying on the two dimensional sphere x P S 2 . Moreover because of divergence free assumption we can work with the vorticity formulation of this equation which can be written as: dw t " ∆w t dt`N pw t qdt`k ÿ i"1 f i dB i t`n ÿ j"1`f j`k`wt g j q dB k`j t , w 0 P L 2 pS 2 , Rq.
( . ) Here pB i t q 1ďiďn`k are mutually independent Brownian motions on R, our noise is a mixture of the additive an linear multiplicative noise. ∆ "´∇˚∇ is the negative Bochner Laplacian on S 2 . The non-linearity N pwq is given by N pwq " Bpw, wq for the symmetric operator Bpv, wq " 1 2 p∇¨pvKwq`∇¨pwKvqq.
Operator K is an operator that reconstructs velocity field from the vorticity: u " Kw "´curl ∆´1w (See more details on the derivation of these equation and their further study in [TW ].) Here the Hilbert space H " L 2 pS 2 , Rq and the interpolation spaces generated by the