EXPONENTIAL ERGODICITY FOR STOCHASTIC BURGERS AND 2D NAVIER-STOKES EQUATIONS

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1 EXPONENTIAL ERGODICITY FOR STOCHASTIC BURGERS AND 2D NAVIER-STOKES EQUATIONS B. GOLDYS AND B. MASLOWSKI Abstract. It is shown that transition measures of the stochastic Navier-Stokes equation in 2D converge exponentially fast to the corresponding invariant measures in the distance of total variation. As a corollary we obtain the existence of spectral gap for a related semigroup obtained by a sort of ground state trasformation. Analogous results are proved for the stochastic Burgers equation. 1. Introduction In this paper we study ergodic behaviour of two important equations arising in Statistical Physics: the stochastic Burgers equation and the stochastic Navier-Stokes equation in 2D. In both cases we assume that the random forcing is correlated in space and white in time. The problem of ergodicity and the rate of convergence to invariant measure in various norms for those two equations was an object of intense research in recent years. In the paper [15] the existence and uniqueness of invariant measure for the stochastic Navier-Stokes equation in 2D was proved in the case when the random force is sufficiently close to the space-time white noise. The exponential rate of convergence of transition measures to the invariant measure µ of the stochastic Navier-Stokes equation was proved for the first time in [1] for µ-almost every initial condition and subsequently for every square integrable initial condition in [22] and [26] see also [1]. In all those papers various versions of coupling technique were applied to prove the convergence properties in metrics equivalent to the topology of weak convergence of measures or an intermediate metric, cf. [26]. The coupling method proved also to be useful to handle random forces which are degenerated in space. For Navier-Stokes equation perurbed by finite-dimensional Wiener process, uniqueness of invariant measure has been proven in a recent paper [18]. Let us note also that similar result were obtained for the forcing consisting of a sequence of random excitations arising in discrete moments of time random kicks, see e.g. [21]. In this work we continue the approach initiated in [15] assuming that the random force is sufficiently nondegenerate. In particular, we are using results of [11], [12], [3] and [9], where the strong Feller property and irreducubility have been proven in an appropriately chosen state space for particular cases of the stochastic Navier-Stokes equation and stochastic Burgers equation. Our main result may be described as follows. Let ut, ζ : t, ζ O} be a solution to either the stochastic Navier-Stokes equation in which case O is a bounded domain in R 2 or a solution to the stochastic Burgers equation and then O =, 1 and let µ be the corresponding invariant measure. Then for any initial distribution ν of the L 2 O-valued Key words and phrases. Navier-Stokes equation, Burgers equation, additive noise, invariant measure, uniform exponential ergodicity, spectral gap. This work was partially supported by the ARC Discovery Grant DP34646, the UNSW Faculty Research Grant PS5345 and the GAČR Grant 21/4/75. 1

2 2 B. GOLDYS AND B. MASLOWSKI random variable u, the probability distribution P t ν of the random variable ut, enjoys the property P t ν µ var P t ν µ V Ce βt ν V, 1.1 where var denotes the norm of total variation of measures and ν V stands for the norm of total variation of the measure V dν considered on L 2 O. The function V : L 2 O [1, in 1.1 is an appropriate Lyapunov function. A class of functions V is found for which 1.1 holds is also provided and shown to include V x = 1 + x p for p > and V x = exp x 2α L 2 L 2 for α, 1. Finally, we derive from 1.1 the spectral gap property of the V -transform P V t P V t φx = V 1 xe V ut, φut, of the semigroup P t. Namely, we show that for φ L p H, V µ, p 1, P t V φ V L 1 p φv dµ V dµ C p e βt/p φ p V dµ L 2 L 2 Exponential convergence to the invariant measure in the distance of total variation and the spectral gap property 1.2 seem to be new for both equations studied in this paper. The main idea of the proof consists in verifying the V -uniform ergodicity for a skeleton process and to this end a geometric drift towards a nontrivial small set must be shown. It is proven that levelsets of the function V are nontrivial small sets and then the corresponding Lyapunov- Foster condition is verified by means of Ito formula. The proof is given for a general Markov process taking values in a Polish space and satisfying conditions H1-H4 cf. Section 3 and then these abstract conditions are verified for Markov processes defined by stochastic Navier-Stokes and Burgers equations, respectively, under suitable assumptions on the correlation of the noise. It may be expected that these assumptions are not optimal, it should be possible to find other sets of conditions implying H1-H4 in particular cases. The general scheme of the proof of V -uniform ergodicity of the skeleton exploits wellknown results from the theory of Markov chains cf. [23], similar idea was used in [27] for some stochastic semilinear equations. For stochastic reaction-diffusion equations V -uniform ergodicity has been proven recently by a slightly different method that allows to give some explicit bounds on the convergence rate [17]. It may be interesting to note that we also obtain an independent proof of existence of invariant measure for both Navier-Stokes and Burgers equation it is an easy consequence of 3.2 and 3.9, cf. [24] or [25] for similar results, which however is known in both cases. The paper is divided into five sections including the Introduction. In Section 2 we provide a rigorous framework for our results and formulate the main theorems, separately for stochastic Navier-Stokes and Burgers equations the Section is divided into two parts. In the case of Navier-Stokes equation only the Dirichlet type boundary conditions are studied in detail, although the case of periodic boundary conditions may be treated as well; only minor changes in the proofs are needed and the noise may be even more degenerate Hypothesis 2.23 and Theorem In Section 3 we study V -uniform ergodicity for a general Markov process in in a Polish space. These results are applied in Sections 4 and 5 to the stochastic Navier-Stokes and Burgers equations, respectively. In order to avoid clumsy notation, the same symbols may sometimes have different meaning in different sections. All results concerning the Navier-Stokes and Burgers equations are given in separate sections, so there is no risk of confusion. For example, H and X t are the state space and solution to the Navier-Stokes equation in Sections 2.1 and 4 or the Burgers

3 equation in Sections 2.2 and 5. The symbols P t and Pt denote the Markov semigroups of a general Markov process taking values in a Polish space E in Section 3 and the general Definition 2.4, or of the Markov processes defined by the Navier-Stokes equation Sections 2.1 and 4 or the Burgers equation Sections 2.2 and 5. The notation is explained at the beginning of particular sections. 3 ACKNOWLEDGEMENT. The authors would like to thank to Franco Flandoli for his valuable remarks. 2. Main Results 2.1. Navier-Stokes Equation. In this Section the equation u t, ζ ν ut, ζ+ut, ζ ut, ζ+ pt, ζ = fζ+ηt, ζ, t, ζ, O, 2.1 t is considered, where ut, ζ = u 1 t, ζ, u 2 t, ζ and pt, ζ denote the velocity and pressure fields respectively, of a viscous incompressible fluid in a bounded domain O R 2 with a smooth boundary O, f is a deterministic external force, η is a random forcing of white noise type and ν > denotes the viscosity. Incompressibility condition reads and initial and the Dirichlet boundary conditions and div ut, ζ =, t, ζ [, O, 2.2 u, ζ = u ζ, ζ O, 2.3 ut, ζ = for t, ζ [, O, 2.4 are considered. We will study an abstract version of system and its weak solution cf. [13]. Set H = x L 2 O } 2 : div x =, x n O =, and V = x H 1 O 2 : div x = }, where n is the outward normal to O cf. [28] for an interpretation of the condition x n =. Identifying H 1 with a subspace of V the dual space of V we have V H V and if there is no danger of confusion, stands for the pairing between V and V. Furthermore, define a closed operator A in H by the formula Ax = νπ x, doma = H 2 O 2 V, where Π is the orthogonal projection of L 2 O 2 onto H. The space V coincides with dom A 1/2 and is endowed with the norm x V = A 1/2 x. The operator A is strictly positive, selfadjoint and its resolvent is compact. For β > we will denote by H β the domain of fractional power A β equipped with the norm x β = A β x. The bilinear operator B : V V V is defined as Bu, v, z = zζ uζ vζdζ, u, v, z V. 2.5 O

4 4 B. GOLDYS AND B. MASLOWSKI Then we may rewrite system in the abstract form dxt + AX t + B X t, X t dt = fdt + GdW t, t, X = x. In the equation above X t is identified with ut, and x with u. The noise in 2.6 is modelled as a standard cylindrical Wiener process W t on H defined on a stochastic basis Ω, F, F t, P, f H and G is a bounded injective linear operator on H. Following [13] we adopt a definition of solution to equation 2.6 resembling the classical definition of a weak solution as understood in the theory of deterministic PDE s. 2.6 Definition 2.1. A progressively measurable process X t = Xt x is a generalised solution to equation 2.6 if X x C[, T ]; H L 2 [, T ]; H 1/4 P-a.s. and X t, y + t X s, Ay ds t P-a.s. for all x H, t [, T ] and y doma. Remark 2.2. By the incompressibility condition we obtain B X s, y, X s ds = x, y + t f, y + W t, G y, Bu, v, z = Bu, z, v, u, v, z V, 2.7 and by the Sobolev embedding theorem there exists a universal constant C such that which justifies Definition 2.1. Assume that Bu, v, u C v V u 2 L 4 O C v V u 2 1/4, img dom A 1 +ɛ 4, 2.8 for some ɛ > and consider an Ornstein-Uhlenbeck process defined by the equation dzt + AZ t dt = GdW t, Z =. 2.9 It is well known, see e.g. [8] that under assumption 2.8 equation 2.9 has a unique progressively measurable mild solution Z t taking values in dom A 1/4 P-a.s. The following result has been proven in [13]. Proposition 2.3. Assume 2.8. Then for each initial condition x H there exists a unique solution X x to equation 2.6, which additionally enjoys the property X x Z L 2 [, T ]; V P a.s. 2.1 Moreover, the family of solutions X x : x H} forms a Markov family which satisfies the Feller property and has an invariant measure µ NS. Our next aim is to define the concept of V -uniform ergodicity of a Markov semigroup. The definition is formulated for a general Markov process X t with values in a Polish space E. Let bb denote the space of bounded Borel functions on E and let P t, P t and P t, x, denote the Markov semigroup on bb, the adjoint Markov semigroup on the space P of probability measures on E, and the transition probability measures, respectively, associated to the process X x t. More precisely, for any t P t : bb bb, P t φx = E x φ X t, 2.11

5 and P t φx = E φyp t, x, dy, φ bb, x E, 2.12 Pt : P P, Pt ν = E P t, x, νdx, 2.13 where E x denotes the expectation corresponding to the initial condition X = x. Obviously, Pt ν may be interpreted as the probability distribution of X t, ν being the initial distribution. Let V : E [1, be a measurable function and let b V B denote the space of Borel functions on E endowed with the norm φ V = sup x E φx V x. Definition 2.4. The Markov semigroup P t is said to be V -uniformly ergodic if P t extends to operator P t : b V B b V B and there exist C > and ω > such that where µ P is the invariant measure. sup P t φx φ, µ CV xe ωt, t x E, 2.14 φ V 1 Let ν var denote the norm of total variation of a signed measure ν and let ν V denote the so-called V -variation: ν V = sup ν, φ = V ν var, 2.15 φ V 1 where we denoted by V ν the measure dρ = V dν. Obviously ν var ν V. In terms of the adjoint Markov semigroup 2.14 implies P t ν µ var P t ν µ V C ν V e ωt, t, ν P Note that one can have ν V = ν, V =. In the rest of Section 2.1 P t and P t will denote the Markov semigroups defined by the solution of the Navier-Stokes equation 2.6 on the space E = H. Under suitable nondegeneracy conditions on G it has been proven cf. [15], [11] that P t ν µ NS as t in the metric of total variation of measures for each initial measure ν P. Building upon these results we aim at proving the V -uniform ergodicity under suitable assumptions on G and V. Hypothesis 2.5. The operator G is Hilbert-Schmidt on H and there exist α 1 4, 1 2 and ɛ > such that dom A 2α img dom A α +ɛ The second inclusion in 2.17 is slightly stronger than 2.8 because we need more regularity of the solution to 2.6. The first inclusion is a nondegeneracy condition. Hypothesis 2.6. The function V : H [1, is measurable and where Φ C 2 R +, Φ 1, Φ, c 1 Φ x 2 V x c 2 Φ x 2, x H, 2.18 lim Φr =, r and for any α, k > there exist β, C > such that αrφr + k r Φ r + r 1/2 Φ r βφr + C,

6 6 B. GOLDYS AND B. MASLOWSKI for r > large enough. Example 2.7. It is easy to see that the functions and satisfy conditions of Hypothesis 2.6. V x = 1 + x p, p >, 2.2 V x = e x 2α, α, 1, 2.21 Remark 2.8. i Without loss of generality we may suppose that 2.19 holds for all r > changing perhaps the constant C. ii If two functions U and V satisfy Hypothesis 2.6 then αu + βv and αv βx satisfy this hypothesis for any α, β > as well. The following is our main result on the stochastic Navier-Stokes equation with the Dirichlet boundary conditions. Theorem 2.9. Let the operator G and the function V satisfy Hypotheses 2.5 and 2.6 respectively. Then the Markov semigroup associated to the Navier-Stokes equation 2.6 is V -uniformly ergodic, i.e and 2.16 hold true with E = H and µ = µ NS, where P t and Pt denote the Markov semigroups defined by the equation 2.6. In particular, V may be defined by 2.2 or We will denote by φ p the norm φ p p = H φx p V xµ NS dx of a function φ in the space L p H, V µ NS. Let 1 V P tv φx = 1 V x P tv φx. Theorem 2.1. The family of operators 1 V P tv defines a C -semigroup on L p H, V µ NS for p 1, and V µ NS is an invariant measure for this semigroup. Moreover, there exist β, C p > such that 1 V P tv φ V µ NS, φ 1 V C P e βt/p φ p, φ L p H, V µ NS p Remark Stochastic Navier-Stokes equation with periodic boundary conditions Using the results from the B.Ferrario s paper [12] it is possible to make similar conclusions in case when the Dirichlet boundary conditions 2.4 are replaced by periodic ones. It may be interesting that in such case the noise may be more degenerate. More specifically, consider the problem , where O =, L 1, L 2, endowed with the periodic boundary conditions ut, ξ + L i η i = ut, ξ, t >, ξ R 2, i = 1, 2, 2.23 where η 1, η 2 is the canonical basis of R 2 and L i is the period in the ithe direction. The system , 2.23, may be formalized in terms of the abstract equation of the form

7 2.6 see e.g. [12] for details, where the state spaces are defined by means of the space H m p O 2 of functions from H m loc R2 2 with zero average and the period L 1, L 2, e.g. H = u H po 2 ; div u = }, V = u H 1 po 2 ; div u = }. Then A = ν with doma = H 2 po 2 H and the spaces H α are defined again as domains of fractional powers of the operator A equipped with the graph norm. The Wiener process W t is standard cylindrical on H and G is a bounded injective linear operator on H. The following is our main condition on the noise term: Hypothesis There exist α > 1 and ɛ > such that doma α img doma α 2 +ɛ The existence and uniqueness of solutions, the Markov property and existence of an invariant measure may be shown as in Proposition 2.3, cf. [13] and [12]. Our result on Navier-Stokes equation with periodic boundary conditions is formulated as follows: Theorem Let f H α 2 1 and assume that Hypotheses 2.6 and 2.12 are satisfied. Then 2 the conclusions of Theorem 2.9 V -uniform ergodicity and Theorem 2.1 the spectral gap hold true for the Markov semigroups P t and Pt defined by the system , Stochastic Burgers Equation. In this section we study the stochastic Burgers equation u t t, ζ ν 2 u ζ 2 t, ζ = 1 u 2 t, ζ + ηt, ζ, t, ζ,, 1, ζ with viscosity ν >, the Dirichlet boundary conditions and the initial condition ut, = ut, 1 =, t, 2.26 u, ζ = u ζ, ζ [, 1] Following a well known approach e.g. [5], [6], [9], we will rewrite system as an evolution equation dxt + AX t dt = 1 2 D ζ X 2 t dt + GdWt, 2.28 X = x, in the space H = L 2, 1, where X t = X x t is identified with ut, and x with u H. In equation 2.28, W t stands for a standard cylindrical Wiener process in H defined on a stochastic basis Ω, F, F t, P, G H H is a bounded operator, D ζ is the distributional derivative operator and A = ν 2 ζ 2, doma = H2, 1 H 1, 1. Denote by e ta a symmetric C -semigroup generated by A in H. Similarly as in the previous Section we define the Ornstein-Uhlenbeck process Z t = t e t sa GdW s, t. 2.29

8 8 B. GOLDYS AND B. MASLOWSKI It is well known that Z C [, T ]; C, 1 P-a.s. cf. p. 14 of [8]. The difference Yt x = Xt x Z t satisfies formally the equation d dt Y t x + AYt x = 1 2 D ζ Y x t + Z t 2, t >, Y x = x. 2.3 We will define Yt x as a solution to the integral equation Y x t = e ta x t e t sa D ζ Y x s + Z s 2 ds, t Definition A process X x C[, T ]; H is said to be a mild solution of equation 2.28 if and only if Y x = X x Z a solution to equation Proposition cf. [9], pp. 262 and 273 For any x H there exists a unique mild solution to equation 2.28 and there exists an invariant measure µ B for the Markov semigroup associated to equation Hypothesis The operator Q = GG has the following properties: trq < and for some δ 1 2, 1 im A δ/2 im Q 1/ Under the nondegeneracy condition 2.32 Da Prato and Debussche proved in [3] that the Markov semigroup associated to 2.28 is strongly Feller in H. Irreducibility has been proven for Q = I by Da Prato and G atarek in [6] cf. also [9], it is however easy to adapt their proof to the present case see Proposition 5.1 for a sketch of the proof. Therefore, for each initial measure ν P the probability distributions P t ν converge to µ B, as t, in the norm of total variation. We shall prove a stronger result. Theorem Let the operator Q and the function V satisfy Hypotheses 2.32 and 2.6, respectively. Then the V -uniform ergodicity 2.14 and 2.16 holds true, where the transition semigroup P t and its adjoint P t are associated to equation 2.28, E = H and µ = µ B. In particular, V may be defined as in 2.2 or We will denote by φ p the norm φ p p = H φx p V xµ B dx of a function φ in the space L p H, V µ B. Let 1 V P tv φx = 1 V x P tv φx. Theorem The family of operators 1 V P tv defines a C -semigroup on L p H, V µ B for all p 1, and V µ B is an invariant measure for this semigroup. Moreover, there exist β, C p > such that 1 V P tv φ V µ B, φ 1 V Ce βt/p φ p, φ L 2 H, V µ B p

9 3. Results on Markov Processes In this section some results on V -uniform ergodicity and L p -ergodicity are stated for general time homogeneous Markov processes. These results are applied to Markov processes defined by the stochastic Navier-Stokes and Burgers equations in the next sections. Throughout the present section we assume that X t a time-homogeneous Markov process in a Polish space E and P t, P t and P t, x, are the respective Markov semigroup, its adjoint and the transition kernel as defined in We also assume that there exists an invariant measure µ P: P t µ = µ, t. The following hypotheses are supposed to be satisfied, where V : E [1, is a measurable function. H1 P t, x, : t >, x E} is a family of equivalent measures. H2 There exists a measurable subspace E 1 E such that for each t > and Γ B the mapping x P t, x, Γ is continuous in E 1. H3 For each r > 1 there exist T > and a compact K E 1 such that where V r = y E : V y r}. H4 For certain k, α, c > inf P T, x, K >, 3.1 x V r E x V X t kv xe αt + c, t. 3.2 Theorem 3.1. Assume H1-H4. Then the Markov semigroup P t is V -uniformly ergodic, i.e and 2.16 hold true. Proof. Take t > such that ke αt 1 4. By H4 we have for t t, x E. Therefore, Taking r 4c we find that and thereby E x V X t V x ke αt 1 V x + c 3 V x + c, E x V X t V x c, x E, t t V x + c < 1 4 r + c, for x E \ V r, E x V X t V x 1 2 V x + ci V r x, x E, t t. 3.5 The last inequality implies that each skeleton chain X nt n with t t has a geometric drift toward V r. We will show that there exists a skeleton for which V r is a nontrivial small set. By H1 each skeleton X nτ, τ >, is ψ-irreducible where ψ = P 1, x, and x E is arbitrary and fixed. Hence, cf. Lemma 2 in [19] or Theorem in [23] there exists a small set Π B, ψπ >, that is and P 1, x, Π >, 3.6 inf P T, x, Γ λγ, Γ B, 3.7 x Π 9

10 1 B. GOLDYS AND B. MASLOWSKI for some T > and a nonnegative measure λ such that λπ >. By the Chapman- Kolmogorov equation we have inf P 2T + T, x, Γ inf P T, y, Γ P T + T, x, dy x V r x V r Π λγ inf P T + T, x, Π λγ inf P T, y, ΠP T, x, dy, 3.8 x V r x V r K where t and K are given in H3. By H1 and 3.6 the function y P T, y, Π is positive on E and by H2 it is continuous with respect to the topology of E 1, in which K is compact. Therefore, by H3 inf P 2T + T, x, Γ δ 1 λγ inf P T, x, K δ 2 λγ, Γ B, 3.9 x V r x V r for some δ 1, δ 2 >. It follows that V r is a nontrivial small set for each skeleton X nm2t +T where m 1. Taking m large enough so that τ = m2t + T t we obtain the skeleton X nτ which is V -uniformly ergodic, i.e. sup P nτ φx µ, φ C e nτω V x, x E, n, 3.1 φ V 1 for some C, ω >. Therefore, by H4 sup φ V 1 P nτ+s φx µ, φ sup φ V 1 P s P nτ φ µ, φ x C e nτω E x V X s C e nτω kv xe αs + c C V xe nτ+sω e ωs ke αs + c CV xe nτ+sω, x E, s [, τ], 3.11 for some C >, which completes the proof. The following simple lemma will be useful to verify condition H1. Lemma 3.2. Let E 1 E with continuous and dense embedding and suppose that the process X t is strongly Feller with respect to the topology of E 1 i.e. H2 holds and let X t be E 1 -topologically irreducible i.e. P t, x, U > for each t >, x E 1 and nonempty open U E 1 and P t, x, E 1 = 1 for all x E and t >. Then condition H1 holds as well, i.e. the measures P t, x, are equivalent for all t > and x E. Proof. The proof based on a modification of an earlier result by Khasminskii cf. [2] is given in [15] for a special choice of spaces E and E 1 but it can be easily extended to the present case. In the last part of this Section we will consider the problem of existence of the spectral gap in the weighted space L p E, V µ, p 1,, with the norm φ p p = φx p V xµdx, φ L p E, V µ. E For any Radon measure ν on E and g L 1 E, ν we denote by gν the measure gνb = gxνdx, B BE. Let 1 V P tv φx = 1 V x P tv φx. B n,

11 11 If 2.14 holds then 1 V P tv φ 1 V µ, φ V Ce ωt φ. We will denote by Q t the semigroup of bounded operators Q t = 1 V P tv : bb bb. Let us recall that if the semigroup P t satisfies the condition H1 then the spaces L p E, µ are invariant for P t and P t φx = p t x, yφyµdy E where p t x, the the density dp t, x, /dµ. The next lemma is essentially known. It collects the facts necessary to prove Theorem 3.4. Lemma 3.3. Assume that H1-H4 are satisfied. Then the following holds. a The measure V µ is invariant for the positive semigroup Q t. b Let ν = ψv µ with ψ L 1 E, V µ. Then where G t ψy = with p t, given by Moreover, Q t ν = G t ψ V µ, E p t x, yψxµdx 3.13 Q t var = G t L 1 E,V µ L 1 E,V µ. c The space L p E, V µ is invariant for the semigroup Q t for each p [1, ] and sup Q t L p E,V µ L p E,V µ <. t Moreover, Q t is a C -semigroup on L p E, V µ for p [1, and for p > 1 it may be, in fact, identified as the dual of the extension of Q t : L q E, V µ L q E, V µ for q = p p 1. Proof. a Clearly, Q t is a positive semigroup on M b E with the invariant measure V µ: 1 V P tv V µ = V Pt 1 V µ = V µ. V b For ψ bb, ψ, we define G t ψy = E p t x, yψxµdx, where G t ψ is a well defined measurable function. Let ν = ψv µ. For φ L E, V µ such that φ 3.13 and the Fubini Theorem yield V P t V 1 ψv µ, φ = ψµ, P t V φ = p t x, yv yφyµdyψxµdx = p t x, yψxµdx φyv yµdy E E E E = G t ψ V µ, φ. 3.14

12 12 B. GOLDYS AND B. MASLOWSKI Hence 3.14 holds for arbitrary ψ L 1 E, V µ and φ L E, V µ and V P t V 1 ψv µ = G t ψ V µ. Moreover, Since = sup φ 1 G t ψ V µ var = sup G t ψ V µ, φ φ 1 1 ψv µ, V P tv φ C T ψ 1, t [, T ]. G t ψ V µ, φ = L 1 G t ψ, φ L, we find that G t is bounded on L 1 E, V µ and is fact the restriction of V Pt 1 V from M be to this space and 3.13 holds. b If ψ L E, V µ is nonnegative then for f from L 1 E, V µ the arguments similar as in a imply V P t 1 ψv µ, f = G t ψ V µ, f C T ψ V f 1, t [, T ], since 1 V P tv 1 1 C T, and therefore V Pt V 1 ψ L E, V µ. All those arguments extend immediately to an arbitrary ψ L 1 E, V µ and b follows by a standard density argument. c This part follows again by the density argument. Theorem 3.4. Assume H1-H4. Then the family of operators 1 V P tv defines a C - semigroup on L p E, V µ, p 1, with the invariant measure V µ. Moreover, there exist β, C p > such that 1 V P tv φ V µ, φ 1 V C p e βt/p φ p, φ L p E, V µ p Proof. By Theorem 3.1 and Lemma holds and therefore By Lemma 3.3 we find that and Therefore, by interpolation we obtain Q t ν V µ var Ce ωt ν var. Q t φ µ, φ 1 Ce ωt φ 1 Q t φ µ, φ C φ. Q t φ µ, φ p C p e ωt/p φ p, for all p 1,. Finally by c of Lemma 3.3 Q tφ V µ, φ 1 V C p e ωt/p φ p, p for each p 1,, which completes the proof.

13 4. Proofs: Stochastic Navier-Stokes Equation The aim of this section is to prove Theorems and 2.13 from Section 2.1. We are going to use the abstract results of Section 3 and to this end we need to verify conditions H1-H4 for the Markov process defined by the stochastic Navier-Stokes equation 2.6 and a function V satisfying Hypothesis 2.6. The full proof is given for the case of Dirichlet boundary conditions Theorems 2.9 and 2.1. The case of periodic boundary conditions Theorem 2.13 may be treated very similarly the sketch of the proof is given at the end of the section. Throughout the section the notation from Section 2.1 is preserved, Hypotheses 2.5 and 2.6 are supposed to hold true, E = H and the Markov semigroups are those defined by the solution to the Navier-Stokes equation 2.6. Lemma 4.1. Let β [, 1 2. Then for any t > there exists a random variable C t depending on t only and such that Xt x 2 β C t x 2 e x 2, P a.s., 4.1 where Xt x denote the solution to equation 2.6 starting from X x = x H. Proof. Recall that Z t is the Ornstein-Uhlenbeck process defined by 2.9. It is well known that the operator A is positive and selfadjoint, with A 1 compact and the eigenvalues α n of A have the property α n n 2. Hence, by the second inclusion of 2.17 Q is of trace class and Z C [, T ], H β, P a.s., 4.2 for each T > cf. [7]. Fix T t and set Yt x = Xt x Z t for x H. Obviously, Yt x satisfies the equation d dt Y t x + AYt x + B Yt x + Z t, Yt x + Z t = f, t >, Y x = x, 4.3 and by Proposition 2.3 Y x C [, T ]; H L 2, T ; V cf. [13], [14] or [29]. By 4.2 it suffices to show 4.1 with Xt x replaced with Yt x. Following arguments in the proof of Proposition 4.1 in [13] we find that for each t [, T ], ɛ > and sufficiently large Cɛ > P-a.s. 1 d 2 dt Y t x 2 + Yt x 1/2 = B Yt x + Z t, Yt x + Z t, Yt x + f, Yt x ɛ Yt x 2 1/2 + ɛ Y t x 2 + Cɛ Z t 4 1/4 Y t x 2 + Z t 4 1/4 + f 2, 4.4 P-a.s. Taking ɛ < 1 and invoking the Gronwall Lemma we obtain Y x t 2 C 1 x 2 e t 1+ Z s 4 1/4ds + t e t τ 1+ Z s 4 1/4ds Z τ 4 1/4 + f 2 dτ for all t [, T ] and a certain universal constant C 1, which together with 4.2 yields 13, 4.5 sup Yt x 2 L x 2, x H, P a.s., 4.6 t [,T ] where L 1 is a finite random variable. Using 4.6, 4.2 and again 4.4 we find that sup t 1 t 2 T t2 t 1 Yt x 2 β dt C 2 sup t 1 t 2 T t2 t 1 Y x t 2 1/2 dt L x 2, x H, P a.s., 4.7

14 14 B. GOLDYS AND B. MASLOWSKI where C 2 is a universal constant and L 2 is a finite random variable. Since B : H θ H ρ H δ is bounded for θ, ρ > and δ [, 1 such that δ + θ + ρ 1 and δ + ρ > 1 2 cf. [16], taking θ = ρ = β 2 and δ = 1 2 β we obtain for v H 1 2 +β and z H β 2 A 1 2 +β Bv + z, v + z, A 1 2 +β A 1 v C 4 + β v + z A 2 +β v C v β + z β v 1 +β, where C stands for a universal constant which may be different on each line. Therefore, by interpolation A 1 2 +β Bv + z, v + z, A 1 2 +β v C v β v 1/2 v 1 2 +β + z 2 1 v 4 + β β ɛ v Cɛ v 2 +β β v 2 1/2 + z 4 1, β 2 for each ɛ > and each constant Cɛ depending on ɛ only. By classical arguments see e.g. [13] we find that Y x L 2, T ; H λ for each λ < 1, which yields for t, T 1 d 2 dt Y t x 2 β + Y t x β = A 1 2 +β B Yt x + Z t, Yt x ɛ Yt x β + Cɛ Yt x 2 β Y t x 2 1/2 + Z t 4 1 and therefore for < s t T Yt x 2 x β C Y and in virtue of 4.7 s 2 β exp t s Y x r 2 1/2 dr + C t Yt x 2 x β C Ys 2 β el 21+ x 2 + Ce L 21+ x 2 s + Z t + f, A 1 2 +β Y x t 4 + β 2 + f 2, 4.1 t Z r β exp Yτ x 2 1/2 dτ dr, r t s Z r β dr 2 L 3 Y x s 2 β e x 2, 4.12 where L 3 is a finite random variable independent of x H and < s t T. Integrating 4.12 with respect to s [ 1 2 t, t ] and invoking 4.7 we obtain Y x t 2 β 2 t L 3 e x 2 t t /2 Y x s 2 β ds 2 t L 3 Ce x 2 t t /2 Y x s 2 β ds L 4 x 2 e x 2, 4.13 for a random variable L 4 depending on t only, which together with 4.2 completes the proof. Theorem 4.2. Assume that the function V satisfies Hypothesis 2.6. Then H4 holds. Proof. Let e n : n 1} be the orthonormal basis of H consisting of the eigenvectors of A and let Π m be the orthogonal projection onto H n = lin e k : k n}. Set, for n 1 and B n x = Π n B Π n x, Π n x, x n = Π n x, x H, G n = Π n GΠ n, f n = Π n f.

15 We will consider the finite dimensional equations dxn t + AX n tdt + B n X n t dt = f n dt + G n dw t, t >, X n = x n H n 4.14 Without the loss of generality we may assume that V x = Φ x 2. Denote by τ R the exit time of X n from the ball B R = y H : y < R}. For a fixed λ > the Ito formula yields EV X n t τ R e λt τ R V x n + E t τr [ λe λs V X n s +e λs 2Φ X n s 2 AX n s, X n s + 2Φ X n s 2 B n X n s + f n, X n s +2 Φ X n s 2 Xn s 2 tr G n G n + Φ X n s 2 tr G n G n]ds V x n + E t τr λe λs V X n s +e λs κφ X n s 2 X n s 2 + k Φ X n s 2 X n s 2 + Φ X n s 2 X n s ds, for any t > and some κ, k >. Hypothesis 2.6 implies that EV X n t τ R e λt τ R t τr V x n + E λe λs V X n s + e λs βv X n s + C ds, 4.16 for some β, C >. Taking λ, β we have EV X n t τ R e λt τ R V x n + E For R we obtain by the Fatou Lemma: t τr e λs Cds V x n + C λ eλt EV X n t V x n e λt + C λ It remains to justify passing with n. Set for m = 1, 2,... Φr if r m, Φ m r = Φm if r m. Clearly, V m y V y as m for each y H and thereby EV m Xt EV X t, m By [2] X n t X t for each t in distribution on the space H δ for each δ > and therefore Moreover, by 4.18 EV m X n t EV m X t, n. 4.2 EV m X n t EV X n t V x n e λt + C λ,

16 16 B. GOLDYS AND B. MASLOWSKI and V x n V x so 4.19 and 4.2 yield EV X t V xe λt + C, λ t Proof of Theorems 2.9 and 2.1. We need to verify conditions H1-H4 for the Markov process defined by equation 2.6. Under Hypothesis 2.5 it has been proven in [11] that the Markov semigroup P t is strongly Feller and irreducible in the space E 1 = H α. Furthermore, since α < 1 2, 4.2 and 2.1 imply that P t, x, E 1 = 1 for x E = H and t >. Hence by Lemma 3.2 conditions H1 and H2 are satisfied with the above choice of spaces E and E 1. Condition H4 has been verified in Proposition 4.2 and it remains to check condition H3. Let β α, 1 2 and for R > put K R = z H β : z β R} By compactness of the operator A 1 each K R is relatively compact in E 1 = H α. Since Φr for r, for each T > there exists L > such that inf P T, x, K R inf P T, x, K R 1 sup P T, x, H \ K R x V r x L x L = 1 sup P Xt x 2 β R2, 4.22 x L hence by Lemma 4.1 inf P T, x, K R 1 P C T x 2 e x 2 R 2 x V r 1 P C T R2 e L2 L 2 for R sufficiently large which completes the proof of H3. >, 4.23 Proof of Theorem The proof almost exactly follows the lines of preceding one, however, the verification of our general condition H1-H3 is based on the results from [12], Theorem 3.2, where it was shown that the Markov semigroup is strongly Feller and irreducible in the space E 1 = H α. Also, it is 2 standard to check that P t, x, E 1 = 1 for all x H and t >, so Hypotheses H1 and H2 are satisfied. Condition H3 may be verified as in Lemma 4.1 by induction cf. Proposition 3.1 in [12] for a similar proof. Finally, condition H4 follows from Proposition 4.2, which applies to the present case without change note that the second inclusion in Hypothesis 2.23 implies that the operator G is Hilbert-Schmidt on H. 5. Proofs: Stochastic Burgers Equation In this section Theorems 2.17 and 2.18 are proven. By Lemma 3.2 it suffices to show that conditions H1-H4 are satisfied for the Markov process defined by the Burgers equation 2.28 and a function V satisfying Hypothesis 2.6. Throughout this section we preserve the notation from Section 2.2 and we assume that Hypotheses 2.16 and 2.6 hold true. Proposition 5.1. The Markov semigroup defined by equation 2.28 is strongly Feller and irreducible.

17 Proof. The strong Feller property has been proven in [3]. Irreducibility was shown in [6] cf. also [9] for Q = I, the proof may be however, easily adapted to the case of more general Q. For the reader s convenience we provide a sketch of the proof. Let żu = Az u + Q 1/2 u, z u 5.1 = x. where u L 2, T ; H. The system 5.1 is approximately controllable which follows from the fact that imq 1/2 = H. That is, for any x, y H and ɛ > we can find u L 2, T ; H such that z u T y < ɛ. Assume now that x, y H 1. Then zu C, T ; H 1 and B z u C, T ; H. Let 17 and ψt = B z u t + Q 1/2 ut. ψ n t = nq 1/2 I + nq 1/2 1 B z u t + Q 1/2 ut = Q 1/2 φ n t Clearly, ψ, ψ n, φ n L 2, T ; H and ψ n ψ in L 1, T ; H. Let us rewrite equation 5.1 in the form żu = Az u + B z u + ψ, z u = x, and consider also the equation żu n = Az u n + B z u n + Q 1/2 φ n, z u n = x. It is known that the corresponding integral equation z u nt = S t x + t S t s B z u ns ds + has a unique solution z u n C, T ; H 1. We claim that t S t s Q 1/2 φ n sds, 5.2 z u n z u in C, T ; H It follows that for any ɛ we can find n big enough, such that zn u = x and znt u y < ɛ. Since H 1 is dense in H we find that the last estimate can be obtained for any y H. Then using the same arguments as in [9] we can prove that that for any t >, y H and r > P t I z H; z y <r} x >, x H If x / H 1 then we can find a sequence x n H 1 such that x n x and then by the result in [3] we find that E X x t X xn t 2 uniformly in t [, T ]. Therefore, 5.4 holds for any x H and the irreducibility follows. Lemma 5.2. Let β, 1 4. then for each t > there exists a random variable C t depending on t only and such that X x t β C t 1 + x 2, P a.s., 5.5 where X x t denotes the solution starting from X x = x H.

18 18 B. GOLDYS AND B. MASLOWSKI Proof. the proof is based on a priori estimates given in Chapter 14 of [9]. for Yt x = Xt x Z t where Z t is defined in 2.29 we have for a fixed T t t t t Yt x 2 x 2 exp 8 Z s 2 ds + 2 Z s 4 8 exp Z r 2 dr ds, t [, T ], x H, s 5.6 where stands for the norm in L, 1. Since trq < it follows that for each λ < 1 2 Z C [, T ]; H λ C [, T ]; C, 1, 5.7 and by 5.6 there exists a random variable C 1 such that Yt x 2 C x 2, P a.s., x H. 5.8 Furthermore, by Lemma of [9] there exists a constant C β > such that t e t sa D ζ usds 2 β C β sup us L 1,1, t T, 5.9 s t for β, 1 4 and u C [, T ]; L 1, 1 and by the analyticity of the semigroup e ta it follows that Xt x β Yt x β + Z t β e ta x β + 1 t e t sa D ζ Y x s + Z s ds 2 + Z t β β c 1 t β x C β sup Yt x + Z t 2 L + Z t β s t 1,1 c 1 t β x + C β sup Ys x 2 + Z s 2 + Z t β, t T, s t for a universal constant c 1 >, which together with 5.7 and 5.8 yields 5.5. Proposition 5.3. Let the function V satisfy Hypothesis 2.6. then H4 holds true. Proof. Following [3] we consider a sequence of approximating equations dxn t + AX n tdt = B n X n t dt + Π n GdW t, X n = x n = Π n x, where Π n are defined as orthogonal projections on the spans of the first n eigenvectors of the operator A and B n x = Π n D ζ Π n x 2, x H 1. Taking into account that B n x, x = and Xn x X x in L 2 Ω, F, P; Hcf. [3] we may repeat word by word the proof of Proposition 4.2 i.e. apply the Ito formula to the process t e λt V Xnt x with λ > small enough, estimate the drift term using Hypothesis 2.6 and pass with n. Proof of Theorem We have to verify conditions H1-H4 for the Markov process defined by equation 2.28 with E = E 1 = H. By Lemma 3.2 and Proposition 5.1 H1 and H2 hold true while H4 follows from Proposition 5.3. It remains to verify H3. To this end we proceed in a similar way as in the proof of Theorem 2.9. For a fixed β, 1 4 set } K R = z H β : z β R, R >.

19 Obviously, K R is relatively compact in E 1 = H for each R > and for each r > there exists L > such that so by Lemma 5.2 inf P T, x, K R inf P T, x, K R 1 sup P T, x, H \ K R, 5.1 x V r x L x L inf P T, x, K R 1 sup P C T 1 + x 2 R x V r x L 1 P C T which is positive for R large enough and H3 holds. R 1 + L 2 References 19, 5.11 [1] Bricmont J., Kupiainen A. and Lefevere R.: Exponential mixing of the 2D stochastic Navier-Stokes dynamics, Comm. Math. Phys , [2] Capinski M. and Gatarek D.: Stochastic equations in Hilbert space with applicaton to Navier-Stokes equations in Any dimension, J.Funct. Anal , [3] Da Prato G. and Debussche A.: Differentiability of the transition semigroup of the stochastic Burgers equation, and application to the corresponding Hamilton-Jacobi equation, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 9 Mat. Appl , [4] Da Prato G. and Debussche A.: Maximal dissipativity of the Dirichlet operator corresponding to the Burgers equation, in Stochastic processes, physics and geometry: new interplays, I Leipzig, 1999, 85-98, CMS Conf. Proc., 28, Amer. Math. Soc., Providence, RI, 2 [5] Da Prato G., Debussche A. and Temam R.: Stochastic Burgers equation, NoDEA Nonlinear Differential Equations Appl , [6] Da Prato G. and Gatarek D.: Stochastic Burgers equation with correlated noise. Stochastics Stochastics Rep , [7] Da Prato G., Kwapien S. and Zabczyk J.: Regularity of solutions of linear stochastic equations in Hilbert spaces, Stochastics , 1-23 [8] Da Prato G. and Zabczyk J.: Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge 1992 [9] Da Prato G. and Zabczyk J.: Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge 1996 [1] E W., Mattingly J. C. and Sinai Ya.: Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation, Comm. Math. Phys , [11] Ferrario B.: Ergodic results for stochastic Navier-Stokes equation, Stochastics Stochastics Rep [12] Ferrario B.: Stochastic Navier-Stokes equations: Analysis of the noise to have a unique invariant measure, Annali Mat. Pura Appl , [13] Flandoli F.: Dissipativity and invariant measures for stochastic Navier-Stokes equations, NoDEA , [14] Flandoli F. and G atarek D.: Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields , [15] Flandoli F. and Maslowski B.: Ergodicity of the 2-D Navier-Stokes equation under random perturbations, Comm. Math. Phys , [16] Giga Y. and Miyakawa T.: Solution in L r of the Navier-Stokes initial value problem, Arch.Rational Mech. Anal , [17] Goldys B. and Maslowski B.: Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE s, preprint of the School of Mathematics, The University of New South Wales, 24, [18] Hairer M. and Mattingly J.C.: Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, preprint 24,

20 2 B. GOLDYS AND B. MASLOWSKI [19] Jain N. and Jamison B.: Contributions to Doeblin s theory of Markov processes Z. Wahrscheinlichkeitstheorie Verw. Geb , 19-4 [2] Khasminskii R.Z.: Ergodic properties of recurrent diffusions and stabilization of the Cauchy problem for parabolic equations, Teoriya Veroyat. Primen , 7-28 in Russian [21] Kuksin S. and Shirikyan A.: A coupling approach to randomly forced nonlinear PDE s I, Comm. Math. Phys , [22] Kuksin S. and Shirikyan A.: Coupling approach to white-forced nonlinear PDE s, J. Math. Pures Appl , no. 6, [23] Meyn S.P. and Tweedie R.L.: Markov Chains and Stochastic Stability, Springer-Verlag, Berlin 1993 [24] Maslowski B. and Simao I.: Asymptotic properties of stochastic semilinear equations by the method of lower measures, Colloquium Math , [25] Maslowski B. and Simao I.: Long-time behaviour of non-autonomous SPDE s, Stochastic Processes Appl., 95 21, [26] Mattingly J. C., Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics, Comm. Math. Phys , [27] Shardlow T.: Geometric ergodicity for stochastic PDE s, Stochastic Anal. Appl., , [28] Temam R.: Navier-Stokes Equations, North-Holland, Amsterdam, 1984 [29] Vishik M.J. and Fursikov A.V.: Mathematical Problems of Statistical Hydromechanics, Kluwer, Dordrecht, 198 School of Mathematics, The University of New South Wales, Sydney 252, Australia address: B.Goldys@unsw.edu.au Mathematical Institute, Academy of Sciences of Czech Republic, Žitná 25, Prague 1, Czech Republic address: maslow@math.cas.cz