LARGE DEVIATIONS FOR STOCHASTIC TAMED 3D NAVIER-STOKES EQUATIONS. (2) 0 g N (r) 2/(ν 1), r 0. {W k

Transcription

1 LAGE DEVIATIONS FO STOCHASTIC TAMED 3D NAVIE-STOKES EQUATIONS MICHAEL ÖCKNE, TUSHENG ZHANG, XICHENG ZHANG Abstract. In this paper, using weak convergence method, we prove a large deviation principle of Freidlin-Wentzell type for the stochastic tamed 3D Navier-Stokes equations driven by multiplicative noise, which was investigated in [ Introduction Let D = 3 or T 3 =three dimensional torus. Consider the following stochastic tamed 3D Navier-Stokes equation in D: [ dut = ν ut ut ut + pt g N ut 2 ut dt [ + ft, utdt + p k t + h k t, ut dwt k, 1 k=1 divut =, u = u. Here, denotes the Euclidean norm on 3, u = u 1, u 2, u 3 is the velocity field, pt, x and p k t, x are unknown scalar functions. The function g N : + + is smooth and satisfies g N r =, if r N, g N r = r N/ν, if r N + 1, 2 g N r 2/ν 1, r. {W k t ; t, k = 1, 2, } is a sequence of independent one dimensional standard Brownian motions on some complete filtered probability space Ω, F, P ; F t t. The stochastic integral is understood as Itô s integral, and the coefficients t, x, u ft, x, u 3, t, x, u ht, x, u 3 l 2, satisfy assumptions H1 and H2 specified below. Here l 2 denotes the Hilbert space consisting of all sequences of square summable real numbers with standard norm l 2. It was proven in [21 that under H1 and H2 below, there exists a unique strong solution for equation 1 in the sense of PDE and SDE. The ergodicity of equation 1 was also proved in [21. By virtue of the presence of the taming function g N, equation 1 can be considered as a modified stochastic Navier-Stokes equation. Without the term g N, equation 1 is the stochastic 3D Navier-Stokes equation for which it is well known that there exists a martingale solution cf. [15, 13. But the uniqueness is open, and one only knows the existence of a locally unique strong solution cf. [16, 27. Moreover, without the term g N, the existence and ergodicity of invariant measures for equation 1 have already been studied by using Kolmogorov operators cf. [1, 8, 9 and the existence of Markovian selections cf. [12, 13. We would like to emphasize that the deterministic tamed equation has the following feature: if there exists a bounded solution to the classical 3D Navier-Stokes equation, then this solution also solves the tamed equation. 1

2 The purpose of this paper is to study the small noise large deviation for the stochastic tamed 3D Navier-Stokes equation 1. The large deviations for stochastic partial differential equations SPDE have been studied by many authors. For example, a large deviation principle LDP for stochastic reaction diffusion equations with non-lipschitz reaction term was established by Cerrai and öckner in [7. An LDP for a Burgers -type SPDE was considered by Cardon-Weber in [6. A uniform LDP for parabolic SPDE was proved by Chenal and Millet in [5. An LDP for stochastic reaction diffusion equations was established by Sowers in [23. A small time large deviation principle for stochastic parabolic equations was obtained by the second named author in [24. For the general theory of large deviations, the reader is referred to the monograph [1. Because of the different nature of the nonlinearities for different types of equations, the large deviations for SPDEs has to be dealt with on individual bases. To establish the large deviation principle for the above stochastic tamed Navier-Stokes equation, we shall adopt the weak convergence method developed by Dupuis-Ellis in [11, 2, 3. This method has been proved to be very effective for various types of stochastic equations cf. [4, 18, 19, 26, 14, 27. The main advantage of this method is that one avoids the use of the usual complicated time discretization and the proof of exponential tightness. Our proof is inspired by Liu s method [14 and also works for 2D stochastic Navier-Stokes equations. This paper is organized as follows: Section 2 contains preliminaries and the formulation of our main result. In Section 3, we shall prove the main result by the weak convergence method. 2. Preliminary and Main esult Throughout this paper, we fix T > and assume ν = 1. We shall use the following convention: The letter C with or without subscripts will denote a positive constant, whose value may change from one place to another An abstract criterion for the Laplace principle. It is well known that there exists a Hilbert space U so that l 2 U with Hilbert-Schmidt embedding operator J and {W k t, k N} is a Brownian motion with values in U, whose covariance operator is given by Q = J J. For example, one can take U as the completion of l 2 with respect to the norm generated by the scalar product h, h U := k=1 h k h k k 2, h, h l 2. For a Polish space B, we denote by BB the Borel σ-field, and by C T B the continuous function space from [, T to B, which is endowed with the uniform distance so that C T B is again a Polish space. Define with the norm H := { h = } ḣsds : ḣ L 2, T ; l 2 T 1/2 h H := ḣs 2 l 2ds, where the dot denotes the weak derivative. Let µ be the law of the Brownian motion W in C T U with covariance space l 2. Then C T U, H, µ 2 3

3 forms an abstract Wiener space. For T, M >, set D M := {h H : h H M} and { h : [, T l 2 is a continuous and F t -adapted A T M := process, and for almost all ω, We equip D M with the weak topology in H. Then h, ω D M }. 4 D M is a compact Polish space. 5 Let S be a Polish space and let I : S [, be given. Definition 2.1. The function I is called a rate function if for every a <, the set {f S : If a} is compact in S. Let {Z : C T U S,, 1} be a family of measurable mappings. Assume that there is a measurable map Z : H S such that LD 1 For any M >, if a family {h,, 1} A T M as random variables in D M converges in distribution to h A T M, then for some subsequence k, Z k + h k k converges in distribution to Z h in S. LD 2 For any M >, Z : D M S is weakly continuous. Equivalently, if {h n, n N} D M weakly converges to h H, then for some subsequence h nk, Z h nk converges to Z h in S. For each f S, define If := 1 2 inf h 2 H, 6 {h H : f=z h} where inf = by convention. Then under LD 2, If obviously is a rate function. We recall the following result from [3 see also [28, Theorem 4.4. Theorem 2.2. Under LD 1 and LD 2, {Z,, 1} satisfies the Laplace principle with the rate function If given by 6. More precisely, for each real bounded continuous function g on S: lim log E µ [ exp gz = inf{gf + If}, 7 f S where E µ denotes expectation with respect to µ. In particular, the family of {Z,, 1} satisfies the large deviation principle in S, BS with rate function If. More precisely, let ν be the law of Z in S, BS, then for any A BS inf If lim inf f Ao log ν A lim log ν A inf If, f Ā where the closure and the interior are taken in S, and If is defined by Statement of the Main esult. In order to formulate our result, we need to introduce some functional analytic framework. To simplify the notations we simultaneously use D to denote the whole space 3 or the three dimensional torus T 3. We assume that the reader can distinguish these two cases below. Let C D; 3 denote the set of all smooth functions from D to 3 with compact ports. For p 1, let L p D; 3 be the vector valued L p -space in which the norm is 3

4 denoted by L p. For α, let H α be the usual Sobolev space on D with values in 3, i.e., the closure of C D; 3 with respect to the norm: 1/2 u H α = I α/2 u dx 2. D Here as usual, I α/2 is defined by Fourier transform. For two separable Hilbert spaces K and H, L 2 K; H will denote the space of all Hilbert-Schmidt operators from K to H with norm L2 K;H. Set for α H α := {u H α : divu = }, 8 where the divergence is taken in the sense of Schwartz distributions. Then H α, H α is a separable Hilbert space. We shall denote the norm H α in H α by H α. We remark that H is a closed linear subspace of the Hilbert space L 2 3 ; 3 = H. Let P be the orthogonal projection from L 2 D; 3 to H. Since we are considering the torus or the full space, it is well known that P commutes with the derivative operators, and that P can be restricted to a bounded linear operator from H m to H m. For any u H 2, define and for k N Au := P u Pu u Pg N u 2 u 9 Bt, u := Ph k t, u k N. 1 Letting the operator P act on both sides of equation 1, system 1 can be written as the following equivalent abstract stochastic evolution equation: [ dut = Aut + Pft, ut dt + Bt, utdw t, 11 u = u H 1. Consider the following small perturbation of equation 11: [ du t = Au t + Pft, u t dt + Bt, u tdw t, u = u H 1, 12 and assume that H1 There exist a constant C T,f > and a function H f t, x L 1 [, T D such that for any t [, T, x D, u 3 and j = 1, 2, 3 x jft, x, u 2 + ft, x, u 2 C T,f u 2 + H f t, x, u jft, x, u C T,f. H2 There exist a constant C T,h > and a function H h t, x L 1 [, T D such that for any t [, T, x D, u 3 and i, j = 1, 2, 3 ht, x, u 2 l 2 + x jht, x, u 2 l x jht, x, u 2 l 2 C T,h u 2 + H h t, x and u jht, x, u l 2 + x j u iht, x, u 2 l u j u iht, x, u l 2 C T,h. 4

5 It was proved in [21, 2 that under H1 and H2, there exists a unique functional such that u t, ω := Φ W, ωt satisfies and u t = u + Φ : C T U C T H 1 13 u, ω L 2, T ; H 2 for P -almost all ω Ω 14 Our main result is the following: [ Au s + Pfs, u s ds + Bs, u sdw s. Theorem 2.3. Assume H1 and H2 hold. Then {u,, 1} satisfies the large deviation principle in C T H 1 L 2, T ; H 2 with rate function I given by If := 1 2 where u h solves the following equation: u h t = u + inf h 2 H, 15 {h H : f=u h } [Au h + Pfs, u h ds + Bs, u h tḣsds. 16 emark 2.4. The existence and uniqueness of strong solutions in the sense of PDE of equation 16 can be proved as done in [ Some Estimates. For any u, v H 2, write Au, v := Au, I v H. 17 We first recall the following result cf. [21, Lemma 2.3. For the reader s convenience, a proof is provided here. Lemma 2.5. For any u H 2, we have and Au H C1 + u 6 H + u 2 H2 18 Au, u 1 2 u 2 H u u 2 L + 2 N u 2 H + u 2 H. 19 Proof. First of all, by Gagliado-Nirenberge s inequality and Young s inequality, we have which gives the first assertion. For inequality 19, we have and by Young s inequality Au H u H 2 + C u L u H + C u 3 L 6 u H 2 + C u 3/4 H u 5/4 H + C u 3/2 2 H u 3/2 H 2 C1 + u 2 H + 2 u 6 H. P u, I u H = I u 2 H + u, I u H = u 2 H 2 + u 2 H + u 2 H, Pu u, I u H 1 2 I u 2 H u u 2 L u 2 H u u 2 L 2, 5

6 where Noting that we have 3 3 u 2 = u k 2, u 2 = i u k 2. k=1 k,i=1 g N u 2 u 2 N and g N u 2, 2 Pg N u 2 u, I u H = g N u 2 u, u L 2 g N u 2 u, u L 2 3 = i u k i g N u 2 u k dx u 2 g N u 2 dx k,i= i u k g N u 2 i u k g N u 2 i u 2 u k dx k,i=1 3 = u 2 g N u 2 dx 1 g 2 N u 2 u 2 2 dx 3 3 u 2 u 2 dx + N u 2 H. 3 Combining the above calculations yields 19. We also need the following lemmas. Lemma 2.6. For any u, v H 2 and α 3/2, 2, we have Au Av, u v 1 2 u v 2 H + C u 2 v 2 H 1 + α u 4 H + 1 v 4 H1. 21 Proof. By 17 and ab 1 4 a2 + b 2, we have Au Av, u v = u v, I u v H u u v v, I u v L 2 g N u 2 u g N v 2 v, I u v L u v 2 H 2 + u v 2 H 1 + u u v v 2 L 2 + g N u 2 u g N v 2 v 2 L 2. By Hölder s inequality and Sobolev s embedding theorem we have, for any α 3/2, 2 u u v v 2 L 2 2 u v 2 L u 2 L v 2 L 6 u v 2 L 3 and by g N r 2 ν = 1 C u v 2 H α u 2 H 1 + C v 2 H 1 u v 2 H α g N u 2 u g N v 2 v 2 L 2 C u v 2 L 6 u 2 + v 2 2 L 3 C u v 2 H 1 u 4 H + 1 v 4 H 1. Combining the above calculations, we obtain 21. 6

7 Lemma 2.7. There exists a constant C > such that for every u H 2 and t [, T Pft, u 2 H C u 2 H + H ft L 1 D, 22 Bt, u 2 L 2 l 2 ;H C u 2 H + H ht L 1 D, 23 Bt, u 2 L 2 l 2 ;H 1 C u 2 H 1 + H ht L 1 D, 24 Bt, u 2 L 2 l 2 ;H 2 C u 2 H 2 + u 4 H 1 + H ht L 1 D. 25 Proof. We only prove the last one. The others are easier. Since P is a bounded linear operator from H m to H m, by H2 we have Bt, u 2 L 2 l 2 ;H 2 C ht, u 2 L 2 l 2 ;L ht, u 2 L 2 l 2 ;L 2 where = x1, x2, x3. Noting that by H2 we have C u 2 H + H ht L 1 D + 2 ht, u 2 L 2 l 2 ;L 2, 2 ht, u = 2 xht, u + 2 x u iht, u u i + u iht, u 2 u i + 2 u i u jht, u ui u j, 2 ht, u 2 L 2 l 2 ;L 2 C u 2 H + H ht L 1 D + u 2 H 1 + u 2 H 2 + u 4 H 1 C u 2 H 2 + H ht L 1 D + u 4 H now follows by combining the above calculations. The following lemma is a direct consequence of H1 and H2. Lemma 2.8. There exists a constant C > such that for any u, v H and t [, T Pft, u ft, v H C u v H, 26 Bt, u Bt, v L2 l 2 ;H C u v H Proof of Theorem 2.3 For proving Theorem 2.3, the main task is to verify LD 1 and LD 2 for S := C T H 1 L 2, T ; H 2, Z := Φ W,, 1, Z h := u h, h H. Let h A T M converge as in probability to h AT M as random variables in H. Set u t, ω := Φ W, ω + 1 h, ω t, 28 where Φ is given by 13. Thanks to h A T M, i.e., T h s, ω 2 l ds M, by Girsanov s 2 theorem, u solves the following control equation: [ u t = u + Au s + Pfs, u s ds + Bs, u sḣsds + We first prove the following uniform estimate. 7 Bs, u sdw s. 29

8 Lemma 3.1. There exists a positive constant C T,M,N,u > such that for any [, 1 E u t 2 H + 1 t [,T T Here u t := u h t, where u h t solves equation 16. E u s 2 H 2ds C T,M,N,u. 3 Proof. We only prove 3 for u t with, 1. The case of u t := u h t can be proved similarly. Below, the constant C will be independent of. Consider the evolution triple H 2 H 1 H. By Lemmas 2.5, 2.7 and 14 we know T Au s H ds + T Thus, by Itô s formula cf. [17 we have Bs, u s 2 L 2 l 2 ;H 1 ds < +, P u t 2 H = u 2 1 H + 2 Au s, u sds For I 1, by 19 we have Bs, u sḣs, u s H 1ds + Bs, u sdw s, u s H 1 a.s.. fs, u s, u s H 1ds Bs, u s 2 L 2 l 2 ;H 1 ds =: u 2 H 1 + I 1t + I 2t + I 3t + I 4t + I 5t. 31 I 1t +N u s 2 H 2ds u s u s 2 L 2ds For I 2, by 22 and Young s inequality we have I 2t 2 C u s 2 H ds + 2 u s 2 H ds. fs, u s 2 H ds [ u s 2H + 1 ds u s 2 H 2ds u s 2 H 2ds. For I3, recalling h D M, by 24 and Young s inequality we have [ I3t 2 Bs, u s L2 l 2 ;H 1 ḣs l 2 u s H 1 ds 1 4 1/2 2M [ Bs, u s 2L2l 2;H 1 u s 2H ds 1 1/2 2M u s H 1 Bs, u s 2 L 2 l 2 ;H 1 ds s [,t u s 2 H 1 s [,t + C M 8 [ u s 2H ds.

9 Define for > τ := inf { t [, T : Bs, u s 2 L 2 l 2 ;H 1 ds }. For I5, by Burkholder s inequality and Young s inequality, we similarly have τ E I5s 2 CE [ Bs, u s 2L2l 2;H 1 u s 2H ds 1 s [,t τ Set 1 4 E gt := E Combining the above calculations we get gt + E τ s [,t τ u s 2 H 1 s [,t τ u s 2 H 1 u s 2 H 2ds C u 2 H 1 + C M,NE which implies by Gronwall s inequality that E s [,t τ u s 2 H 1 C u 2 H 1 + C M,N + E T τ Since lim τ = T, by Fatou s lemma, we obtain 3. Let u h solve equation 16. Set w t := 1/2 τ + CE [ u s 2H + 1 ds. 1. τ [ u s 2H ds gs + 1ds, u s 2 H2ds C. Bs, u h sḣs ḣsds. 32 Lemma 3.2. w converges in probability to zero in C T H 2 as. Proof. Let {e k, k N} be an orthonormal basis of H 2 and Π n the projection from H 2 to H 2 n :=span{e 1, e 2,, e n }. Define w n t := Π n Bs, u h sḣs ḣsds. Then by Hölder s inequality and since h, h A T M, we have E w n t w t 2 H 2 t [,T T 2 E I Π n Bs, u h sḣs ḣs H 2ds T 2 E I Π n Bs, u h s L2 l 2 ;H 2 ḣs ḣs l 2ds T 2M 2 E I Π n Bs, u h s 2 L 2 l 2 ;H 2 ds. 9

10 By the dominated convergence theorem, 25 and 3with =, we get lim n,1 Fix n N. We have that E H h w n t w t 2 H = t [,T Π n Bs, u h s hsds by 25 and 3with = is P a.s. a continuous linear map on H for all t [, T with values in H 2 n. Hence, if h k is a sequence of random variables with values in H, P -a.s. weakly converging to H, then Clearly, by 25 and 3with = Π n Bs, u h s h k sds k, P a.s.. 34 [, T t Π n Bs, u h s h k sds, k N are equi-continuous, P a.s.. Hence the convergence in 34 is uniform in t [, T, i.e., takes place in C T H 2. Since a sequence of random variables in a metric space converges in probability if and only if any of its subsequences has a subsequence which P a.s. converges to the same limit point, from the above we conclude that, since h converges in probability to h as random variables in H, w n converges in probability to zero in C T H 2 as. Note that for any η > P w t H 2 2η t [,T P +P t [,T The result now follows by combining this with 33. We now prove: w n t w t H 2 η w n t H 2 η t [,T Lemma 3.3. u defined in 28 converges in probability to u := u h, defined by equation 16, in S = C T H 1 L 2, T ; H 2 as. Proof. Set v := u u h. Then v t = + Au s Au h sds + Pfs, u s fs, u h sds Bs, u sḣs Bs, u h sḣsds + As in Lemma 3.1, by Itô s formula we have v t 2 H = 2 Au s Au 1 h s, v sds +2 Pfs, u s fs, u h s, v s H 1ds 1. Bs, u sdw s. 35

11 and Bs, u s Bs, u h sḣs, v s H 1ds Bs, u h sḣs ḣs, v s H 1ds Bs, u sdw s, v s H 1 Bs, u s 2 L 2 l 2 ;H 1 ds =: I 1t + I 2t + I 3t + I 4t + I 5t + I 6t. Set for > and [, 1 } θ := inf {t : u t 2H1 + u s 2 H 2ds > Then, by 3 we have P θ < T = P E τ := θ θ. t [,T [ u t 2H1 + u s 2 H 2ds > T u t 2 H + u s 2 1 H 2ds / t [,T and C T,M,N,u For I 1, by 21 we have P τ < T C T,M,N,u I 1t τ 3 4 τ τ v s 2 H 2ds + C v s 2 H 2ds + C τ τ v s 2 H αds v s 2 H ds, where we have used Young s inequality and the following interpolation inequality v 2 H C α v α α H 2 v 2 α H, α, 2. For I2, by Young s inequality and 26 we have I 2t τ 1 4 For I 3, by 27 we have I 3t τ C 1 4 τ τ τ τ v s 2 H 2ds + C v s 2 H ds. v s H ḣs l 2 v s H 2ds τ v s 2 H 2ds + C ḣs 2 l 2 v s 2 H ds. 11

12 Hence I 1t τ + I 2t τ + I 3t τ C τ τ v s 2 H 2ds 1 + ḣs 2 l 2 v s 2 Hds. 37 For I5, by Doob s maximal inequality and 24 we have T τ E I5t τ 2 E Bs, u s 2 L 2 l 2 ;H 1 v s 2 H 1ds t [,T Likewise, for I 6 we have E C T,. 38 I6t τ t [,T C T,. 39 We now deal with the hard term I 4. By 32, 35 and Itô s formula we have For I 41, we have 1 2 I 4t = v t, w t H 1 For I 42, by 18 we have I 42t τ Similarly, we have H 2 w s, Au s Au h s H ds w s, fs, u s fs, u h s H 1ds w s, Bs, u sḣs Bs, u h sḣs H 1 ds w s, Bs, u sdw s H 1 =: I 41t + I 42t + I 43t + I 44t + I 45t. I41t τ 1 4 v t τ 2 H + w t τ 2 1 H1. 4 τ C τ w s H 2 Au s Au h s H ds w s H 2 u s 2 H 2 + u hs 2 H 2 + C ds C T, w s H s [,T I 43t τ C T, For I 44, by 23 and Hölder s inequality we have I44t τ w s H 2 s [,T τ w s H s [,T [ Bs, u s L2 l 2 ;H ḣs l 2 + Bs, u h s L2 l 2 ;H ḣs l 2 ds 12

13 Similar to 38, we have τ 1/2 C T M w s H 2 [ u s 2H + u hs 2H + 1 ds s [,T C T,M, w s H s [,T E Combining 37-43, we obtain where Set Then I45t τ 2 C T,. 44 t [,T 1 2 v t τ 2 H τ v s 2 H 2ds τ C 1 + ḣs 2 l 2 v s 2 H 1ds +C T,M, w s H 2 + ξ, s [,T ξ := I45t τ + I5t τ + I6t τ. t [,T t [,T t [,T g t := v s 2 H + 1 s [,t g t τ C T,M, By Gronwall s inequality we obtain g T τ C T,M, C T,M, s [,T s [,T v s 2 H 2ds. w s H 2 + ξ +C 1 + ḣs 2 l 2 g s τds. w s H 2 + ξ T } exp {C 1 + ḣs 2 l 2ds w s H 2 + ξ expc T + M 2. s [,T Therefore, by Lemma 3.2 and 38, 39 and 44, we have Lastly, note that for any δ >. g T τ in probability as. 45 P g T δ P g T τ δ + P τ < T. The desired convergence now follows from 45 and 36. Proof of Theorem 2.3: Let h be a sequence in A T N which converges in distribution to h. Since D N is compact and the law of W is tight, {h, W } is tight in D N C T U by the definition of tightness. Hence for some sequence k, {h k, W } weakly converges to some probability measure ν on D N C T U. Note that the law of h is just ν, C T U. By Skorohod s representation 13

14 theorem, there are random variables h k, W k, k N and h, W on a probability space Ω, P such that 1 h k, W k a.s. converges to h, W in DN C T U; 2 h k, W k has the same law as hk, W for each k N; 3 The law of { h, W } is ν, and the law of h is the same as h. We remark that in the proof of Lemma 3.3, one can replace h k, W by h k, W k. Thus, using Lemma 3.3, we get Φ k W k + h k u h, in probability, k where Φ k is the strong solution functional given by 13. From this, we get Φ k W + h k u h, in distribution, k hence LD 1 holds. LD 2 can be proved as in Lemma 3.3. The theorem now follows from Theorem 2.2. Acknowledgements: The financial port of the AC Discovery grant DP of Australia and of the BiBoS-esearch Centre and of the DFG through the SFB 71 and the Beijing-Bielefeld International Graduate School Stochastic and eal World Models is gratefully acknowledged. Xicheng Zhang is also grateful to the port of NSFNo of China. eferences [1 V.I. Bogachev and M. öckner: Elliptic equations on infinite dimensional spaces and applications. Probab. Theo. el. Fields, 1221, [2 M. Boué and P. Dupuis: A variational representation for certain functionals of Brownian motion. Ann. of Prob., 1998, Vol. 26, No.4, [3 A. Budhiraja and P. Dupuis: A variational representation for positive functionals of infinite dimensional Brownian motion. Probab. Math. Statist. 2 2, no. 1, Acta Univ. Wratislav. No. 2246, [4 A. Budhiraja, P. Dupuis and V. Maroulas: Large deviations for infinite dimensional stochastic dynamical systems. Ann. of Prob., 3628, [5 F. Chenal and A. Millet: Uniform large deviations for parabolic SPDEs and applications. Stochastic Process. Appl , [6 C. Cardon-Weber: Large deviations for a Burgers -type SPDE. Stochastic processes and their applications, , [7 S. Cerrai and M. öckner: Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-lipschtiz reaction term. Ann. Probab. 32:1 24, [8 G. Da Prato and A. Debussche: Ergodicity for the 3D stochastic Navier-Stokes equations. J. Math. Pures Appl , no. 8, [9 A. Debussche and C. Odasso: Markov solutions for the 3D stochastic Navier-Stokes equations with state dependent noise. J. Evol. Equ., 6 26, no. 2, [1 A. Dembo and A. Zeitouni: Large deviations techniques and applications. Springer-Verlag, Berlin Heidelberg, [11 P. Dupuis and.s. Ellis: A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New-York, [12 F. Flandoli, M. omito: Markov selections for the 3D stochastic Navier-Stokes equations. Probab. Theory elated Fields, 14: [13 B. Goldys, M. öckner, X. Zhang: Martingale Solutions and Markov Selections for Stochastic Partial Differential Equations. Stochastic Processes and their Applications, Volume 119, Issue 5, May 29, Pages

15 [14 W. Liu: Large deviations for stochastic evolution equations with small multiplcative noise. arxiv: v3 [math.p [15. Mikulevicius and B. L. ozovskii: Martingale Problems for Stochastic PDE s, in Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs, Vol. 64, pp , AMS, Providence, [16. Mikulevicius and B.L. ozovskii: Stochastic Navier-Stokes equations for turbulent flows. SIAM J. Math. Anal , no. 5, [17 C. Prévôt and M. öckner: A concise course on stochastic partial differential equations. Lecture Notes in Mathematics, 195. Springer, Berlin, 27. vi+144 pp [18 J. en and X. Zhang: Freidlin-Wentzell s large deviations for homeomorphism flows of non-lipschitz SDEs. Bull. Sci. Math. 2 Serie, Vol 129/8 pp [19 J. en and X. Zhang: Freidlin-Wentzell s Large Deviations for Stochastic Evolution Equations. J. Func. Anal., Vol.254, [2 M. öckner, B. Schmuland and X. Zhang: Yamada-Watanabe Theorem for Stochastic Evolution Equations in Infinite Dimensions. Condensed Matter Physics, Vol.11, No.254, [21 M. öckner, X. Zhang: Stochastic Tamed 3D Navier-Stokes Equations: Existence, Uniqueness and Ergodicity. Probability Theory and elated Fields., Vol. 145, No. 1-2, [22 M. öckner, X. Zhang: Tamed 3D Navier-Stokes Equation: Existence, Uniqueness and egularity. arxiv:math/73254 [23.B. Sowers: Large deviations for a reaction-diffusion equation with non Gaussian perturbations. Ann. Probab., 2:1 1992, [24 T.S.Zhang: On small time asymptotics of diffusions on Hilbert spaces. Ann. Probab. 28:2 2, [25 T.S.Zhang: Large deviations for stochastic nonlinear beam equations. J. of Functional Analysis, 248:1 27, [26 X. Zhang: Euler Schemes and Large Deviations for Stochastic Volterra Equations with Singular Kernels. J. of Differential Equation, Vol. 224, [27 X. Zhang: Stochastic Volterra Equations in Banach Spaces and Stochastic Partial Differential Equations. arxiv.org/abs/ [28 X. Zhang: A variational representation for random functionals on abstract Wiener spaces. J. Math. Kyoto Univ., in press. Michael öckner: Fakultät für Mathematik, Universität Bielefeld, Postfach 1131, D-3351 Bielefeld, Germany, Departments of Mathematics and Statistics, Purdue University, W. Laffayette, IN, 4797, USA, roeckner@math.uni-bielefeld.de Tusheng Zhang: School of Mathematics, University of Manchester, Oxford, oad, Manchester M13 9PL, England, U.K., tzhang@maths.man.ac.uk Xicheng Zhang: School of Mathematics and Statistics, The University of New South Wales, Sydney, 252, Australia, Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 4374, P..China, XichengZhang@gmail.com 15