Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces

The main goal of this paper is to 
 generalize to Banach spaces the well-known results for diffusions on Hilbert spaces obtained in [Peszat, S. and Zabczyk, J. (1995). 
 Strong Feller property and irreducibility for diffusions on Hilbert spaces. 
 Ann. Probab. 23(1): 157-172.]. More precisely, we are aiming to prove the strong Feller property and irreducibility of the solutions to a stochastic evolution equations (SEEs) in Banach spaces. We give sufficient conditions on the path space and the coefficients of the SEEs for these aforementioned properties to hold. We apply our result to investigate the long-time behavior of a stochastic nonlinear heat equations on $L^p$-space with $p>4$. Our result implies the uniqueness of the invariant measure, if it exists, for the stochastic nonlinear heat equations on $L^p$-space with $p>4$.

1. Introduction. The uniqueness of an invariant measure for the Markov process associated to the solution of stochastic evolution equations (SEEs) is one of the most important questions in Probability, if not in Mathematics. This question has been very actively investigated in the last thirty years and several important results have been obtained in both finite and infinite dimensional settings. The case of stochastic differential equations (SDEs for short) on the euclidean space R n (or a finite dimensional Riemannian manifolds in more generality) is now quite well understood. Several methods used to prove the uniqueness of invariant measure for SDEs has been successfully extended and developed to the case of stochastic system in infinite dimensional (usually Hilbert) spaces, see, for instance, the new edition of an 1992 monograph by Da Prato and Zabczyk [8], the articles [9], [10] [16] [21] [24] [25] [30] and references therein. One of the most well-known and much used approaches to prove the uniqueness of an invariant measure to a Markov semigroup associated to a SEEs in a Hilbert space consists in proving the strong Feller (SF) property and the irreducibility. Roughly speaking a Markov semigroup P has the SF property if the image, under P, of the space of bounded and measurable real functions defined on the state space is included in the space of bounded and continuous functions. Frequently, for instance if the noise entering the system is nondegenerate, this smoothing property can be proved through the gradient estimate of the semigroup which will follow from the well-known Bismut-Elworthy-Li formula, see, for instance, [8,30]. The study of a SEEs driven by a degenerate noise is very delicate, however, there are now several results treating successfully this case. Recently, two notions of regularity of Markov process which seem very powerful to treat the ergodic property of SEEs in Hilbert spaces driven by degenerate were introduced in the mathematical literature: the asymptotic feller property (ASF), see [13], [14] and the e-property, see [17]. These notions, not present in the finite dimensional literature, are weaker than the SF property but a irreducible Markov semigroup that is asymptotic SF or has the e-property has at most one invariant measure. Very often, at least in the Hilbert spaces setting, one uses the Malliavin calculus to check ASF and the e-property. However it also possible to prove the uniqueness of invariant measure for SEEs in Hilbert space without making use of the Malliavin calculus. For instance, in [19,20] uniqueness of invariant measure for various SEEs in Hilbert spaces with degenerate noise was proved by making use of the coupling method. To the best of our knowledge, the results by Hairer and Mattingly are applicable to systems in Hilbert spaces with degenerate but spatially smooth noise. This paper is the beginning of our effort to study the question of uniqueness of an invariant measure for SEEs driven by both degenerate and spatially irregular noise.
In contrast to the SEEs in Hilbert spaces, there are not so many papers treating the uniqueness of invariant measure of stochastic evolution equations (SEEs) in infinite dimensional Banach spaces. However, we refer to [6], [12], [26] and references therein for few relevant results. It is worth noticing that the method developed in [19], see also [20], can be applied to show the uniqueness of invariant measure of a SEE in Banach space with non-degenerate additive noise. Furthermore, the method elaborated in [19,20] was recently used in [18] to prove that when defined on the space of continuous complex functions the stochastic Complex Ginzburg-Landau with degenerate additive noise has a unique invariant measure. However, the result obtained in [18] relies very much on some good estimates of the solution in the Hilbert-Sobolev spaces H m with m ą n{2, where n is the space dimension and as in the work [14], the additive noise is spatially regular.
The difficulties of the study of the uniqueness of an invariant measure for SEEs in the Banach space setting arise mainly due to the fact that the tools frequently used in the Hilbert space framework cannot be extended in a straightforward way to the Banach space setting. In fact these tools rely very much on some nice properties, which might not hold for general Banach spaces, of the underlying Hilbert spaces. For instance, this is the case of the Bismut-Elworthy-Li formula for SEEs with Lipschitz coefficients whose proof relies very much on the approximation of Lipschitz functions by maps of class C 2 and with bounded derivatives. In contrast to the Hilbert spaces case, such approximation is not always possible for any Banach space. In fact, it is only recently proved in [15] that such approximation is only possible for some classes of Banach space, for instance, Banach spaces that have a norm of class C 2 .
In this paper we are mainly interested in studying the uniqueness of invariant measure, if any, associated to a SEE with multiplicative noise in a martingale type 2 Banach space. We are modestly aiming to extend to the Banach setting the results about SF property and irreducibility of diffusions in Hilbert spaces obtained in [30]. One direction in that quest is the derivation of a Bismut-Elworthy-Li gradient formula for the Markovian semigroup associated to a SEE in a martingale type 2 Banach space with norm of class C 2 . Our main results are stated in Theorem 2.4 and Theorem 2.6 which, roughly speaking, state that the Markov semigroup associated to our SEE is strong Feller and irreducible provided that the coefficient are (in some sense) globally Lipschitz and the state space E is martingale type 2 Banach space with a C 2 -smooth norm. For the proof of these results we closely follow the scheme of the proofs in [30]. As in the Hilbert space setting, the irreducible property is much easier to be proved than the SF property. However, as mentioned earlier, the SF property is difficult to prove and its proof, which consists of two parts, is the subject of Section 3. In the first part of the proof we assume that the coefficients are twice Fréchet differentiable with bounded derivatives. Under this new assumption, by truncating the noise and approximating the (unbounded) linear part of the SEE by sequence of bounded operators (the Yosida approximation), we can prove a Bismut-Elworthy-Li formula. In the second part we exploit the geometry (mainly the smoothness of the norm) of the state space. The C 2 -smoothness of the norm enables us to use recent result in [15] about approximation of Lipschitz functions by sequence of twice differentiable functions which are also Lipschitz continuous. By this construction and a careful passage to the limit we dispense of the differentiability of the coefficients and obtain the SF property of the original problem. Thanks to a Girsanov theorem for SEE in Banach space (see [29]), the proof of Theorem 2.6 is quite easy and use the same argument as the proof of [30,Theorem 1.3]. The driving noise of our equation is non-degenerate and the coefficients of the equations are, roughly speaking, globally Lipschitz continuous. In contrast to [12] (resp. [6]) we do not assume that the state space is densely and continuously embedded in a separable Hilbert space (resp. the coefficients are dissipative).
After we have finished the present paper, we learned that Shamarova [33] used the Malliavin calculus in 2-smooth Banach space to prove that any finite dimensional projection of the law of the solution of a SEE with coefficients satisfying the Hörmander conditions is absolutely continuous with respect to the Lebesgue measure. Although her assumptions are a little bit restrictive, it would be interesting to check whether her approach could be used in our framework to prove the strong Feller property. We should note that the development of Malliavin calculus in Banach spaces is still at its infancy, however, we refer to [22], [23], [32] and [33] amongst others to some significant results that have been obtained. With the help of these results, we hope that in near future we will be able to develop and generalize the approach in [13] and [14], so that we will be able to analyze SEE with degenerate noise in Banach setting.
The organization of this paper is as follows. Section 2 is devoted to the introduction of frequently used notations and hypotheses in this paper. We also state in Section 2 our main results. In Section 3 we prove the strong Feller property of our equations. We applied our results to the nonlinear stochastic heat equations in the L p -space setting in Section 4. The appendix is devoted to the proof of some auxiliary results that we could not find in the literature.

2.
Notations and the main results. Let E be a separable Banach space with norm |¨| . Let pH, |¨| H q be a separable Hilbert space such that the embedding 1054 ZDZIS LAW BRZEŹNIAK AND PAUL ANDRÉ RAZAFIMANDIMBY E Ă H is continuous. Let te k u 8 k"1 be a fixed orthonormal basis of H and E˚be the dual space of the separable Banach space E. We use xϕ, uy to denote simultaneously the dual pairing between ϕ P E˚and u P E and the scalar product of ϕ P H and u P H.
The spaces of bounded measurable (resp. continuous) real-valued functions on E are denoted by B b pEq (resp. C b pEq). The supremum norm on B b pEq and C b pEq is denoted by ¨ 0 . The space of all (bounded or unbounded) linear maps from a Banach space Y 1 into another Banach space Y 2 is denoted by LpY 1 , Y 2 q. The Banach space of bounded linear operators from Y 1 into Y 2 is denoted by L pY 1 , Y 2 q and the norm on L pY 1 , Y 2 q is }¨} L pY1,Y2q .
Before we proceed further, let us recall from [11] the notion of C k -smoothness of the norm of a Banach space E.
The functional φ is called the (Fréchet) derivative of f at x P E and denoted by f 1 pxq. The higher order Fréchet derivatives f pkq are defined inductively, see, for instance, [7] for more details. The set of real valued functions on E whose k-th Fréchet derivative (resp. all of whose Fréchet derivatives) is continuous on an open set U Ă E is denoted by C k pU q (resp. C 8 pU q). Following [11, Chapter V, page 184], we say that the Banach space E has a C k -smooth norm if its norm |¨| belongs C k pEz t0uq. Example (Bonic-Frampton's Theorem). For a measure space pO, Σ, µq with positive measure µ, we denote by L p :" L p pO, Σ, µq, p P r1, 8q, the space of equivalence of function classes f : O Ñ R such that (i) If p is even, then the Banach space L p has a C 8 -smooth norm.
(ii) If p is odd, then L p has a C p´1 -smooth norm.
(iii) if p is not an integer, then L p has a C rps -smooth norm, where rps is the integer part of p.
The notion of C k -smooth bump functions is very important in the approximation of functions defined on a Banach space E by a sequence of C k -smooth functions; we recall its definition below.
Definition (Bump functions). A Banach space E admits a C k -smooth bump function on E if there exists a function ϕ P C k pEq with non-empty and bounded support.
One the most important facts implied by the C k -smoothness of the norm of a Banach space E is the existence of a C k -smooth bump on E. In fact, we have the following lemma.
Lemma 2.1. If a Banach space E has a C k -smooth norm, then it admits a C ksmooth bump function.
Proof. The proof is very similar to the proof of [11,Fact I.2.1], hence we just sketch it. Let τ : R Ñ r0, 1s be a C 8 -smooth function such that τ pxq " By assumption and definition, |¨| is C k -smooth Fréchet differentiable on Ez t0u, thus the function τ p|¨|q belongs C k pEq, and since its support is equal to the closed ballBp0, 2q, it is a C k -smooth bump function on E.
Before we continue further, we state the definition of 2-smooth Banach space and Banach space of martingale type 2.
Definition (Martingale type 2 Banach space). Following [36,Definition A.3] and the equivalence between results in [31], we say that a Banach space E is 2-smooth or equivalently a Banach space of martingale type 2 if there exists an equivalent norm defined by the modulus of smoothness of pE, |¨|q satisfying ρ E ptq ď Kt p for all t ą 0 and some K ą 0. It is known from [27] and [36,Lemma A.5] that that Banach space E is 2-smooth or of martingale type 2 if and only if the function E Q x Þ Ñ |x| P R is C 1 -smooth and its Fréchet derivative is globally Lipshictz. Thus, by the same proof of Lemma 2.1, if E is of martingale type 2, then there exists a C 1 -smooth bump function whose derivative is globally Lipschitz.
Let γ be a canonical cylindrical Gaussian distribution on H. We say that a bounded linear operator L : H Ñ E is γ-radonifying if the image γ˝L´1 of γ by L has an extension to a σ-additive measure γ L on E. The set of all γ-radonifying operators from H into E is denoted by RpH, Eq. For L P RpH, Eq, we put L RpH,Eq :" which in view of Fernique theorem is a finite number. Then, RpH, Eq is a separable Banach space under this norm. Let pΩ, F, Pq be a complete probability space equipped with a filtration F :" tF t : t ě 0u satisfying the usual condition. Let tβ k u 8 k"1 be a sequence of independent, real-valued F-Wiener process. We define a cylindrical Wiener process on H by the series For each q ě 2 we denote by N q pEq the set of all E-valued processes u defined on r0, 8qˆΩ such that uˇˇr 0,tsˆΩ is Bpr0, tsqˆF t -measurable and ż t 0 |upsq| q ds ă 8 P-a.s. for all t ě 0.
By M q pEq we denote the subset of N q pEq such that E ż t 0 |upsq| q ds ă 8 for all t ě 0.

ZDZIS LAW BRZEŹNIAK AND PAUL ANDRÉ RAZAFIMANDIMBY
Let x be a E-valued random variable and let us consider the following stochastic evolution equation where A is the infinitesimal generator of a C 0 -semigroup tSptq : t ě 0u on E, the maps F and B act from E into E and from E into L pH, Eq, respectively. On E, A, F , B and tSptq : t ě 0u we make the following standing assumptions. Assumptions.
(I) The Banach space E has an unconditional Schauder basis and a C 2 -smooth norm.
Hq be a map such that there exists σ P p0, 1 2 q for which A´σB : E Ñ RpH, Eq is Lipschitz, i.e, there exists 2 such that for any u, v P E A´σBpuq´A´σBpvq RpH,Eq ď 2 |u´v|.
(VI) There exists a map G : E Ñ L pH, Hq with the following properties.
(a) BpzqGpzq " Id H for any z P E (b) κ :" sup zPE Gpzq L pH,Hq ă 8 . (VII) There exist two sequences of maps B n : E Ñ LpH, Hq and G n : E Ñ L pH, Hq, n P N, satisfying the following conditions. (a) For any n P N and z P E. B n pzqG n pzq " Id H .
Furthermore, there exists κ ą 0 such that for any n P N sup zPE G n pzq L pH,Hq ă κ.
(b) The sequence A´σB n : E Ñ RpH, Eq is uniformly Lipschitz, i.e., there exists 5 such that for any n P N, x, y P E }B n pxq´B n pyq} RpH,Eq ď 5 |x´y|.
(c) The map A´σB n is C 2 -smooth and its first and second Fréchet derivatives are continuous and bounded. (d) We assume that A´σB n p¨q Ñ A´σBp¨q point-wise in RpH, Eq.
In the next definition we give the notion of solution to the problem (2). Definition 2.2. We say that an E-valued process u " tuptq : t ě 0u is a mild solution to (2) iff u P N q pEq and the following integral equation is satisfied for any t ě 0 P-a.s.
We have the following result. The solution has a continuous modification which is still denoted by u and u P L q pΩ, Cpr0, T s; Eqq for any q ě 2, T ě 0. Moreover, the process u has the Markov property.
Since the proof is almost standard, we just sketch it in Appendix A.
In the present paper we denote by upt, xq the mild solution to SEE (2) with initial condition x P E. The Markovian semigroup P t :" tP t ; t ě 0u associated to SEE (2) is defined by P t ψpxq :" rP t ψspxq " Erψpupt, xqs for any t ě 0, ψ P B b pEq and x P E. We recall that P t is Feller (resp., strong Feller) if for any ψ P C b pEq (resp., B b pEq) and t ą 0, The main theorem of this paper is the following Before we proceed to the statement of the second result, we introduce the following definition.
Definition 2.5. The Markov semigroup P t is called irreducible on E if for any t ą 0, x P E and any open set Γ Ă E we have P t 1 Γ pxq ą 0.
The second result of this paper is the irreducibility of the semigroup P t . Theorem 2.6. Let the assumptions (II)-(VI) hold. Then, there exists a constant κ ą 0 such that for any t ą 0, x, y P E and δ ą 0, Since the proof of Theorem 2.6 is almost identical to the proof of [30, Theorem 1.3], we postpone it till Appendix B.
3. Proof of Theorem 2.4: Strong Feller property. The proof consists of two parts. In the first part we will assume that the coefficients F and B are of C 2 class. In the second part we will use some approximation arguments to remove this restrictive condition. Throughout we fix T ą 0 and q ě 2 and we denote by S q the space of all (equivalence classes of) E-valued progressively measurable processes u such that~u~q The space S q endowed with the norm~¨~q is a Banach space.
3.1. SEE with twice differentiable coefficients. In addition to Assumptions (III)-(VI), we assume in this part that the coefficients F and B in the problem (2) satisfies the following two sets of conditions. Assumptions.
In particular, there exists a positive constant C 1 such that The map A´σB is C 2 -smooth and there exists a constant C 3 ą 0 such that for any z, h, k P E. Here the constant 5 ą 0 is the same as in Assumptions (VIIb), respectively.
From here, to simplify notation we set In the next lemma we give a regularity property of the solution up¨, xq when viewed as a map from E into S p .
Lemma 3.1. Let E be a martingale type 2 Banach space. In addition to Assumptions (III)-(VI) let us assume that the Assumptions (A)-(B) are satisfied and ψ P C 2 b pEq. Then, the map E Q x Þ Ñ up¨, xq P S q , where up¨, xq is the solution to the problem (2), is Gateaux differentiable, and its derivative η h :" u x p¨, xq¨h in direction of h P E is the unique mild solution to Moreover, for any q ě 2 there exist positive constants M 0 , M 1 ą 0 and M 2 ą 0 such that E sup sPr0,ts |η h psq| q ě M 0 e qλt |h| q e C1M1` 5M2 , for every .
The proof of the above lemma will be given later.
The key to the proof of Theorem 2.4 is the following proposition which is the version of the celebrated Bismut-Elworthy-Li lemma for Banach spaces. Proposition 1. Let the assumptions of Lemma 3.1 be satisfied. Let vpt, xq, x P E, t ě 0 be the map defined as in (3).
for any t ą 0, x P E, and h P E.
Proof of Proposition 1. For each positive integer n let A n " nApnI´Aq´1 be the Yosida approximation of A and let Q n be the finite dimensinal projection from H onto H n where H n :" spante 1 , . . . , e n u.
Following [1, Section 5] we introduce the H n -valued Wiener process W n :" Q n W defined by We also set F n :" npI`nAq´1F , and B n pxq :" npI`nAq´1Bpxq˝i n , x P E, where i n is the natural embedding i n : H n ãÑ H. Now, let u n p¨, xq be the unique strong solution of the equation du n pt, xq " pA n u n pt, xq`F n pu n pt, xqqqdt`B n pu n pt, xqqdW n ptq, u n p0q " x. (7) Let t ą 0 be fixed. By applying twice Lemma 3.1 we infer that v n pt,¨q " Erψpu n pt,¨qqs is twice Gateaux differentiable, and to avoid ambiguity we denote by B x v n and B 2 xx v n its first and second directional derivative. Thanks to the Itô's formula in [ where for any t ą 0 and x P E We refer to [8,Definition 9.24] for the definition of a strict solution to (8). Now applying Itô's formula, see for e.g. [3, Theorem A.1] or [28,Theorem 74], to the stochastic process Ψpsq " v n pt´s, u n ps, xqq, s P r0, ts, we obtain that x v n pt´s, u n ps, xqqpB n pu n ps, xqqqdW n psq.
Now, let y n be the R-valued stochastic processes defined by y n ptq :" ż t 0 xB x v n pt´s, u n ps, xqq, B n pu n ps, xqqdW n psqy, t ě 0.
For any h P E let z n be R-valued process defined by z n ptq :" where η h is the derivative of up¨, xq at x P E in the direction h P E (see Lemma 3.1). Due to the Itô formula for z n ptqy n ptq we obtain xrB n pu n ps, xqqs˚˝rB x v n pt´s, u n ps, xqqs, e k ŷ xQ n Gpups, xqqη h psq, e k y  ds˙.

From this last identity and the fact that
Erv n pt, xqz n pt, xqs " v n pt, xqErz n pt, xqs " 0, we derive that Erψpu n pt, xqqz n ptqs " E ż t 0 xrB n pups, xqqs˚˝rB x v n pt´s, u n ps, xqqs, Q n Gpups, xqqη h psqyds.

ZDZIS LAW BRZEŹNIAK AND PAUL ANDRÉ RAZAFIMANDIMBY
Hence it follows from the definition of B n that Erψpu n pt, xqqz n ptqs " E ż t 0 xB x v n pt´s, u n ps, xqq, npI`nAq´1Bpu n ps, xqqGpups, xqqη h psqyds.
Since ψ P C 2 b pEq, we easily infer from the above convergence that there exist a subsequence u n 1 such that P-almost surely where z is the R-valued process defined by zptq " Since vpt,¨q P C 2 b pEq for any t ą 0 we deduce from (10) that where B x r¨sphq indicates the directional derivative in the direction h P E. To obtain the third line in the chain of equalities above, we used the fact that P t , t ě 0 is a Markov semigroup and Evpt´s, ups, xqq "EP t´s ψpups, xqq "P s P t´s ψpxq "P t ψpxq.
It follows from (13) that Recalling the definition of zptq we deduce from the last identity that which ends the proof of Proposition 1.
Now we give the promised proof of Lemma 3.1.
Proof of Lemma 3.1. We will check the first part of the lemma for small T ą 0 and repeating the argument given below on rT, 2T s, . . . , will yield the general case. Let us first notice that the problem (4) is a linear stochastic evolution equation with time dependent and random coefficients. We will justify shortly that the problem (4) has a unique mild solution. For this purpose, let F pt, ω, ηq "F 1 pupt, ωqq¨η p Bpt, ω, ηq "B 1 pupt, ωqq¨η, for any t P r0, T s, ω P Ω, and η P E. Due to Assumptions (A)-(B) there exists a constant C 1 ą 0 such that |F pt, ω, η 1 q´F pt, ω, η 2 q| ď C 1 |η 1´η2 | |A´σrBpt, ω, η 1 q´Bpt, ω, η 2 qs| RpH,Eq ď 5 |η 1´η2 | for any t P r0, T s, ω P Ω, and η 1 , η 2 P E. Since F and A´σB are of class C 2 and the process u is progressively measurable, then for any η P E the stochastic processes F p¨,¨, ηq and A´σBp¨,¨, ηq are progressively measurable. Furthermore, the mapsF and A´σ p B are linear w.r.t the third variable. Due to these facts we can apply [1,Theorem 4.3] to infer that the problem (4) has a unique mild solution. Now, we consider the map Λ : EˆS q Ñ S q defined for x P E, u P S q by Λpx, uqptq :" Sptqx`ż t 0 Spt´sqF pupsqqds`ż t 0 Spt´sqBpupsqqdW psq, " Λ 0 pxqptq`Λ 1 puqptq`Λ 2 puqptq, @t P r0, T s.
We have the following three properties of Λ.
(1) There exists a constant positive T such that Λpx, uq´Λpx, vq~q ď 1 2~u´v~q , for any u, v P S q .
The existence of such T is ensured by Appendix A.
(2) The map Λ is Gateaux differentiable with respect to its variables and for all x, y P E, u, v P S q , and t P r0, T s pΛ x px, uq¨yqptq " Sptqy, To prove this claim it is sufficient to check the following items. (i) The map Λ 0 and Λ 1 are Gateaux differentiable and for all x, y P E, u, v P S q , and t P r0, T s pΛ 0 x pxq¨yqptq " Sptqy,  This yields the claim (ii).
(3) For i " 1, 2 the maps Λ i u : S qˆSq Q pu, vq Þ Ñ Λ i u pu, vq P S q is separately continuous.
The separate continuity of Λ 1 u wrt to its arguments can be treated as in [8], so we just study the continuity of Λ 2 u . Fix u, v P S q and let tv n : n P Nu be a sequence converging to v in S q . Owing to [1, Theorem 3.2] we havẽ Thanks to the boundedness of A´σB 1 we derive that Λ 2 u pu, v n q´Λ 2 u pu, vq~q q ď CT~v n´v~q q , from which we easily derive the continuity of Λ u p¨,¨q wrt to the second variable. Now let tu n : n P Nu be a sequence converging to u in S q . Owing again to [1, Theorem 3.2] we obtaiñ Thanks to assumption (V) we can now argue as in [8,Proposition 9.6] to show that lim nÕ8~Λ 2 u pu n , vq´Λ 2 u pu, vq~q q " 0, which implies the continuity of Λ 2 u wrt to its first variable. The items (1)-(3) and [8, Lemma 9.2] imply that on the small interval r0, T s the solution up¨, xq, x P E to the problem (2) is Gateaux differentiable as a map from E into S q and its derivative η h :" u x p¨, xq¨h in the direction of h P E is the mild solution to (4). Repeating this argument on rT, 2T s,. . . yields the proof of the first part of the lemma. Now let us prove the second part of Lemma 3.
Now, we easily conclude from the Gronwall inequality that η h p¨q satisfies (5). This ends the proof of the lemma.
We state and prove the following (important) lemma. 3.2. SEE with globally Lipschitz coefficients. In this part we will get rid of the stringent assumptions (A)-(B). The strategy of the proof is to find sequences pF n q ně1 and pB n q ně1 approximating F and G and satisfying the assumptions (A)-(B). Thanks to Assumption (VII) it is enough to find a sequence pF n q ně1 satisfying the properties we mentioned earlier. Since, by Lemma 2.1, any Banach space having C k -smooth norm admits a C k -smooth bump function, the following lemma, which is given in [15,Theorem H], is sufficient for such aim.
Lemma 3.3. Let k be a positive integer and E be a Banach space having C ksmooth norm and an unconditional Schauder basis. Let Y be a Banach space. Then, there exists a constant C ą 0 such that for any L-Lipschitz map Φ : E Ñ Y and positive integer n there exists a CL-Lipschitz map Φ n P C k pE, Y q for which Φpxq´Φ n pxq} Y ă 1 n . Moreover, }Φ 1 n pxq¨h} Y ď CL|h|, }Φ 2 n pk, hq} Y ďCL|h||k|, for any x, h, k P E and n P N. Remark 1. Note that the differentiability in the above lemma is taken in the Fréchet sense. The Lipschitz constant of Φ n is independent of n. Now, we state the following result.
Proposition 2. Let E be a Banach space satisfying the assumption (I). Then, there exists a constant C 0 ą 0 and a sequence pF n q ně1 approximating F such that |F n pxq´F n pyq| ď C 0 1 |x´y|, for all x, y P E, n P N.
Moreover, the two estimates in Assumption (A) hold with C 1 " C 0 1 where 1 is the positive constant from Assumption (IV).
Proof. This is a direct consequence of Lemma 3.3.
After stating Proposition 2 we are almost ready to prove our main result (see Theorem 2.4). But before we embark on the proof of Theorem 2.4 we state and prove the following lemma.
Lemma 3.4. Let E be a separable Banach space having C 2 -smooth norm and P t be a Markov semigroup on B b pEq. Let c ą 0 and t ą 0 be fixed constants. Then, the following two conditions are equivalent Proof. Since E is separable and has a C 2 -smooth norm it follows from [11, Theorem VIII.3.2] that bounded continuous function on E can be approximated uniformly by functions of class C 2 . Now, we can prove the lemma by using the same idea as in [30,Lemma 2.2]. Now, we are ready to prove our main result.
Proof of Theorem 2.4. Let n P N. Let B n and G n be the sequences of functions given in Assumption (VII) and F n be the sequence from Proposition 2. Let u n p¨, xq for x P E be the unique mild solution to the problem du n ppt, xq " rAu n pt, xq`F n pu n pt, xqqsdt`B n pu n pt, xqqdW ptq, u n pp0q " x, and P pnq t be the Markov semigroup associated to u n . Thanks to Proposition 2 and Theorem A.1 we derive that for any q ě 2 lim nÑ8 E sup sPr0,ts |u n ppt, xq´upt, xq| q " 0.
Let ψ P C 2 b pEq. Thanks to the boundedness and continuity of ψ we derive that as n 1 Ñ 8 P pn 1 q t ψpxq Ñ P t ψpxq for all t ě 0. Proposition 1 is applicable to P pnq t since F n , B n and G n satisfy all of its assumptions. Hence, we can also apply Lemma 3.2 to P pnq t . Then, there exist M 1 ą 0, M 2 ą 0 such that for any ψ P C 2 b pEq, x P E, y P E and t ą 0 where C 0 , 1 and 5 are the positive constants in Proposition 2, Assumption (IV) and (VIIb), respectively. Now letting n 1 Ñ 8 and using Lemma 3.4 yield that for any t ą 0 there exists a positive constant c t such that for any ψ P C b pEq, x P E, This concludes the proof of Theorem 2.4. up¨, 0q " up¨, 1q " 0, (15b) up0,¨q " xp¨q P L p pOq.  We denote by A the realization of the Dirichlet Laplacian ∆ in E with domain DpAq " H 2,p pOq X H 1,p 0 pOq. With these notations the problem (15) can be recast in the following abstract SEE duptq " pAuptq`F puptqqqdt`BpuptqqdW ptq,

Application
up0q " x P E.
4.1. Basic auxiliary results. It is well-known that the operator A is the generator of an analytic semigroup Sptq, t ě 0 on E which has an extension, denoted by the same symbol, on H. This extension is again analytic and its infinitesimal generator, again denoted by A, is the realization of the Dirichlet Laplacian on H. Note that H has an orthonormal basis te k : k " 1, 2 . . .u whose elements are eigenfunctions of A. We denote by tλ k : k " 1, 2, . . .u the corresponding set of eigenvalues.
We prove a result about the mapping B.
for any b PẼ " L q pOq and h P H Proof. Let r " 2q q`2 . Observe that A´σ maps L r into H 2σ,r because A has the BIP property and so DpA σ q " rL r , DpAqs σ , and on the other hand rL r , DpAqs σ Ă H 2σ,2 , see [34]. Thus, since, by assumption, 2σ´1 r ě´1 2 , it follows from the Sobolev embedding H 2σ,r Ă H and the above observation that from which along with the Hölder inequality we derive that q`2 we easily conclude the proof of the proposition from the last inequality.
As a crucial consequence of this proposition we have the following result.
Proposition 4. Let p ą 2 and assume that assumptions (A) hold. Then, there exists σ P p0, 1 2 q such that A´σB maps E into RpH, Eq. Moreover, there exist ą 0 such that }A´σ pBpuqr¨s´Bpvqr¨sq } RpH,Eq ď |u´v| E , for any u, v P E.
For any u P E and h P H let Then G maps E into L pE, Hq and satisfies the following properties 1. For any z P E BpzqGpzq " Id H .
Due to the boundedness from below of bp¨q, it is easy to check that G is a welldefined bounded map from E into L pH; Hq. Moreover, pBpuq˝Gpuqqrhs " bpuqˆ1 bpuq h˙" h, for any u P E and h P H. This ends the proof of Proposition 4. Moreover u has a continuous modification which is still denoted by u and u P L q pΩ, Cpr0, T s; Eqq for any q ě 2.
Proof. It follows from the Bonic-Frampton theorem, Proposition 4 and Assumption (B) that all the assumptions (II)-(V) are satisfied by the problem (16). Thus, Theorem 4.1 is a corollary of Theorem 2.3.

4.2.
Uniqueness of the invariant measure of problem (16). Let C 8 b pRq Ă C 8 pRq be the set of all function f such that it and its derivatives of any order are bounded. Let pb n q nPN Ă C 8 b pRq such that lim nÕ8 |b n pxq´bpxq| " 0, @x P R, and there exists a constant ą 0 for which |b 1 n pxq´b n pyq| ď |x´y| for any n P N and x P R. Furthermore, we require that there exists a constant 9 ą 0 such that for any n P N and x P R |b n pxq| ď 9 .
Remark 2. For each n P N let ψ n : R Ñ r0, 8q be the map defined by where c 1 is chosen so that ş R ψ n pxqdx " 1. The sequence pb n q nPN of maps b n : R Ñ R defined by b n pxq " ż R bpyqψ n px´yqdy, x P R, n P N, satisfies the above conditions. For u P E and v P H we set B n puqrvspξq " pb n˝u qpξqvpξq, ξ P O.
We have the following result.
Proposition 5. If σ ą 1 p then the maps, for n P N, E Q u Þ Ñ A´σB n puq " tH Q h Þ Ñ A´σB n puqrhs :" A´σrb n puqhsu P L pH, Hq are -Lipschitz, of class C 2 in the Fréchet sense and its first and second derivative are bounded. Moreover, A´σB n p¨q converges in L pH, Hq to A´σBp¨q point-wise.
Proof. The proof of the -Lipschitz continuity follows the same line as in proof of Proposition 4, so we omit it.
We prove that A´σB n is of class C 2 and have bounded derivatives. For this aim let u P E, v P E and h P H be arbitrary. Now, setB n puqrhs " A´σrb n puq hs and Ψ v " It is easy to see that B n pu`vqrhs´B n puqrhs´A´σrb 1 n puqpvhqs " A´σrv Ψ v hs. Since b 1 n is bounded, Ψ v h is an element of H and it follows from Proposition 3 that |B n pu`vqrhs´B n puqrhs´A´σrb 1 n puqpvhqs| H " |A´σrb n puq hs| H ď c|Ψ v h| H |v| E . We deduce from this thatB n " A´σB n is of class C 1 and its derivative is defined by E Q v Þ ÑB 1 n puqrzs " tH Q h Þ ÑB 1 n puqrzsh :" A´σrb 1 n puqpvhqsu P L pH, Hq. Furthermore, from the uniform boundedness of b 1 n and Proposition 3 we infer that for any u, v P E and h P H, |A´σrb 1 n puqpvhqs| H ď C |v| E |h| H . Thus, the derivative ofB n is uniformly bounded in the sense that there exists a constant C ą 0 such that for any n P N, u P E and v P E }B 1 n puqrvs} L pE,L pH,Hqq ď C |v| E . For any u P E, v P E let With a similar argument as above, we can show that B 1 n pu`vqrysh´B 1 n puqrysh´A´σrb 2 n puqpyvhqs " A´σry v Φ v hs. for any u P E, v P E, y P E and h P H. Thus, we infer from Proposition 3 that Cauchy-Schwarz's inequality yields |B 1 n pu`vqrysh´B 1 n puqrysh´A´σrb 2 n puqpyvhqs| H ď C|Φ v h| H |y| E |v| E , from which altogether with the boundedness of b 2 n we infer the Fréchet differentiability ofB 1 n . So we have just proved thatB n has a second derivative defined byB for any u, v 1 and v 2 P E. Applying Proposition 3 we deduce that there exists a constant C ą 0 such that for any u P E, v 1 P E and v 2 P E }B 2 n puqrv 1 , v 2 s} L 2 pE,L pH,Hqq ď C|v 1 | E |v 2 | E , where L 2 pE, L pH, Hqq :" L pE, L pE, L pH; Hqqq.
It remains to prove the point-wise convergence of A´σB n to A´σB in L pH, Hq. If σ ą 1 p , then thanks to Proposition 3 there exists a constant C ą 0 such that |A´σrpb n puq´bpuqqhs| H ď C|h| H |b n puq´bpuq| E , for any u P E, h P H and n P N. Since b n pxq Ñ bpxq for any x P R and |b n puq| E ď 9 for any n P N, it follows from Lebesgue Dominated Convergence Theorem that |b n puq´bpuq| E Ñ 0, as n Õ 8. Hence, we easily derive from the estimate (17) that A´σB n point-wise converges to A´σB in L pH, Hq. This ends the proof of our proposition.
The next theorem contains the main result of this section. It is clear that S q equipped with the norm~¨~p is a Banach space. For u P S q , ξ P L q pΩ, F 0 , P; Eq, and t P r0, T s we set Spt´sqF pupsqqds`ż t 0 Spt´sqBpupsqqdW psq.
By assumption, it is clear that if u P S q then Λpuq is progressively measurable. Now, we check that Λ maps S p into itself and it is a strict contraction for small T . For this aims let u, v P S q , ξ P L q pΩ, F 0 , P; Eq and Z :" Λpuq´Λpvq. It is easy to check that Thus, it follows from assumption (V) that ı .
This last estimate implies that Λp¨q maps S q into itself and is a contraction for small T . This fact and standard fixed point argument enable us to conclude the first part of the proof of our theorem. Owing to assumption (V) and [1, Theorem 3.2], the stochastic process Spt´sqBpupsqqdW psq, t P r0, T s, has a continuous modification. Thus, u has a continuous modification. The Markov property of the process u follows from the uniqueness result and [8, Theorem 9.14]. This ends the proof of Theorem 2.3.
In the next theorem we give a convergence result which is of great importance in our analysis.
Theorem A.1. Let F and B two maps satisfying assumptions (IV) and (V). Let pB n q ně1 be a sequence of maps B n : E Ñ LpH, Hq such the following conditions hold.
2. The sequence A´σB n p¨q converges to A´σBp¨q point-wise in RpH, Eq.
Let also pF n q ně1 be a sequence of maps F n : E Ñ E satisfying the following items.
(i) There exists a constant ą 0 such that for any n ě 1, x, y P E |F n pxq´F n pyq| ď |x´y|, |F n pxq| ď p1`|x|q.
(ii) F n pxq Ñ F pxq in E for any x P E. Let x P E be fixed and u be the unique mild solution of the problem (2). Let u n be the unique mild solution to the following problem du n ptq " pAu n ptq`F n pu n ptqqqdt`B n pu n ptqqdW ptq, (18a) Then, for any q ě 2 and T P p0, 8q.
Proof. Notice first that the existence and uniqueness of u n is ensured by Theorem 2.3. Let t P p0, 8q, n ě 1 and set Φ 1 n ptq :" Spt´sqrF n pu n psqq´F pupsqqsds, Spt´sqrB n pu n psqq´BpupsqqsdW psq.
We have where for t P r0, T s, Φ 1,1 n ptq :"ˆż t 0 }Spt´sq} L pE,Eq |F n pu n psqq´F n pupsqq|ds˙q, Φ 1,2 n ptq :"ˆż t 0 }Spt´sq} L pE,Eq |F n pupsqq´F pupsqq|ds˙q, and c ą 0 is constant depending only on q. From the Hölder inequality and the Lipschitz continuity of F n we infer the existence of a positive Cp , tq such that A´σrB n pu n psqq´B n pupsqq q RpH,Eq ds A´σrB n pupsqq´Bpupsqqs q RpH,Eq ds.
From the Lipschitz continuity of A´σB n we infer the existence of a positive constant c 0 " Cpt, q, q such that E sup sPr0,ts A´σrB n pupsqq´Bpupsqqs q RpH,Eq ds c 0 E ż t 0 |u n psq´upsq| q ds.
Hence, we have just showed that we can find a positive constant c 0 such that E sup Since, by assumption, there exists a constant ą 0 such that for any integer n ě 1 A´σB n pupsqq q RpH,Eq ds ď p1`E sup sPr0,ts |upsq| q q, and F n Ñ F (resp. A´σB n Ñ B ) point-wise in E (resp. in RpH, Eq), we can apply the Lebesgue Dominated Convergence Theorem to infer that as n Õ 8 A´σrB n pupsqq´Bpupsqqs q RpH,Eq ds Ñ 0.
Appendix B. Proof of Theorem 2.6. The proof of Theorem 2.6 is very similar to the proof of [30,Theorem 1.3]. It relies very much on the following result. We recall that throughout this section u is the unique solution to the problem (2).
Lemma B.1. Let the assumptions of Theorem 2.6 be satisfied. Let t 1 P p0, T q and let f : rt 1 , T sˆE Ñ E be a bounded and measurable mapping. If Z is the solution of dZpsq " pAZpsq`1 pt1,T q psqf ps, Zpsqqqds`BpZpsqqdW psq, Zp0q " x P E, then for t P r0, T s the laws on pE, BpEqq of upt, xq and Zpt, xq are equivalent.
Proof. Since, by Theorem 2.3, the solution up¨, xq of (2) is continuous in E, thus the idea of the proof is very similar to the second part of [30, Proof of Lemma 3.1].
Here we only outline the steps where their proof and ours differ. First, we assume that F is uniformly bounded. Let f 1 : r0, T sˆE Ñ E be the map defined by f 1 ps, xq :" # f ps, xq if s P pt 1 , T s, 0 otherwise.
It is clear that f 1 is a measurable and bounded map. LetF : r0, T sˆE Ñ H be the map defined bỹ F ps, xq :" Gpxqipf 1 ps, xq´F pxqq, s P r0, T s, x P E, where i is the natural embedding i : E ãÑ H. It follows from the Assumption (VI) and the boundedness of f 1 and F thatF is uniformly bounded, i.e., there exists a constant C ą 0 such that |F pt, xq| H ď C, @t P r0, T s, x P E. Then, by the Girsanov theorem in the Banach space setting, see for e.g. [29, Section 7], we infer that (1) the process M p¨q is a martingale under the measure P, EpM ptqq " 1, PpM pT q ą 0q " 1, Owing to (4), we see that Zpsq "Spsqx`ż This means that Z is the solution to the problem (2) underP. Thanks to this remark and the facts (1)-(4), we can follow the exact same lines as in [30, Proof of Lemma 3.1] to prove that the laws on pE, BpEqq of upt, xq and Zpt, xq, t P r0, T s, are equivalent. Second, we will get rid of the additional assumption on F imposed in the first step. For this purpose, for each real number ą 0, we set It is not difficult to check that for each ą 0, the map F : E Ñ E is Lipschitz continuous and there exists a constant C ą 0 such that |F pxq| ď C , @x P E.
Let ą 0 and u p¨, xq be the E-valued continuous solution to (2) with F replaced by F . Arguing as above we can show that the laws on Cpr0, ts; Eq, t P r0, T s, of u p¨, xq and Zp¨, xq are equivalent. Since for each F " F on the ball Bp0, q, by the pathwise uniqueness of solution the laws on Bp0, q of u pt, xq and upt, xq, t P r0, T s, are identical. This implies the equivalence of laws on pE, Bq of upt, xq and Zpt, xq, t P r0, T s. Now we are ready to prove Theorem 2.6.
Proof of Theorem 2.6. Due to Theorem 2.3 and Lemma B.1, the proof is the same as in [30, Proof of Theorem 1.3] so we omit it.
Conlusion. In this paper we have studied the uniqueness of the invariant measure of stochastic evolution equations in Banach spaces. The main difficulty is the proof of the strong Feller property of the Markov solution which was divided in two parts.
In the first part we assumed that the coefficients F and B are C 2 -smooth and established a Bismut-Elworthy-Li (BEL) formula in the context of Banach spaces, see Proposition 1. This BEL type formula along with some nice estimates on the derivative of the solution with respect to the initial condition will imply the strong Feller property. In the second part we used some approximation argument to remove these restrictive conditions. We mainly use the fact that if the state space E has C 2 -smooth norm then it is possible to approximate Lipschitz functions on E by C 2 -smooth and Lipschitz functions. In the near future we are aiming to remove this smoothness condition imposed on the norm of E. The first step is to consider a 2-smooth Banach space E as in [27].