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Unique Ergodicity in Stochastic Electroconvection

We consider a stochastic electroconvection model describing the nonlinear evolution of a surface charge density in a two-dimensional fluid with additive stochastic forcing. We prove the existence and uniqueness of solutions, we define the corresponding Markov semigroup, and we study its Feller properties. When the noise forces enough modes in phase space, we obtain the uniqueness of the smooth invariant measure for the Markov transition kernels associated with the model.


INTRODUCTION
Electroconvection refers to the dynamics of electrically conducting fluids under the influence of electrical charges. There are many instances of electroconvection in non-Newtonian and Newtonian fluids, including the flow of nematic and smectic suspensions subject to applied voltage. A particularly interesting example [8,42] considers the dynamics of a thin smectic film in an annular region, driven by an imposed voltage at the boundary. In [5] the behavior of the system was investigated mathematically in the absence of stochastic forcing. The model was described in terms of a surface charge density q, an electric field E and a fluid velocity u. The dynamics were confined to a two dimensional domain (T 2 in the present paper). The electric field E was derived from a time independent potential Φ representing the voltage applied at the boundary and a dynamic potential Λ −1 q due to the charge density q, via the relation where Λ −1 denotes the inverse of the square root of the two-dimensional periodic Laplacian Λ = √ −∆.
The current density due to the fluid and the electric field E is 2) and the charge density obeys the continuity equation The fluid velocity u obeys the incompresible Navier-Stokes equation forced by the electrical forces qE and time independent body forces f , where p is the fluid pressure and ν is the kinematic viscosity. The well-posedness and global regularity of the deterministic model (1.1)-(1.4) were obtained in [5] in bounded domains with homogeneous boundary conditions, and the long-time dynamics were investigated in [1] in the two-dimensional torus T 2 .
In this paper we consider the stochastic electroconvection model corresponding to the deterministic model (1.1)-(1.4), dq + ∇ ⋅ (qu − ∇Λ −1 q − ∇Φ)dt =gdW, (1.5) du + u ⋅ ∇udt + ∇pdt − ν∆udt = −q(∇Λ −1 q + ∇Φ)dt + f dt + gdW, (1.6) ∇ ⋅ u = 0 (1.7) forced by time independent noise processesgdW and gdW on T 2 . For simplicity, we assume that ν = 1. We address the global well-posedness of (1.5)-(1.7), the Feller properties of the Markov semigroup associated with (1.5)-(1.7), and the existence, uniqueness and regularity of the invariant measures for the Markov transition kernels associated with the model (1.5)-(1.7). A vast literature treats the well-posedness of stochastic partial differntial equations. Martingale type approaches [2,3,9,17,41] were established to prove the existence and uniqueness of solutions to the two-dimensional stochastic Navier-Stokes equations (NSE). In [37], the authors use a different approach, independent of the pathwise solutions, based on a generalization of the classical Minty-Browder local monotonicity argument [38,39], to establish the well-posedness to the stochastic NSE in bounded and unbounded domains. Global existence and uniqueness of strong pathwise solutions were obtained as well for the two-dimensional [14,22,23] and threedimensional [10,24] stochastic primitive equations. The stochastic electroconvection model (1.5)-(1.7) is nonlocal, nonlinear, with critical dissipation in one equation, and consequently the proof of its global well-posedness is rather technical. Under low regularity assumptions imposed on the noises (namely L 4 forg and H 1 for g), we prove that the system (1.5)-(1.7) has unique global solutions when the deterministic initial charge density is L 4 regular and the deterministic initial velocity is H 1 regular. The existence of solutions is obtained by taking a viscous approximation of (1.5)-(1.7), establishing uniform bounds for the viscous solutions, and using weak convergence. The identification of the drift is highly challenging. The reason is that the nonlinearity qRq is not weakly continuous in the spaces we have control in. The remedy is a coercive estimate (3.17) and use of ideas from [37]. As a consequence of the existence result, we define the Markov transition kernels on L 4 × H 1 and we show that they are Feller in the norm of H − 1 2 × L 2 . If the noises have higher regularity (namely ∇g ∈ L 8 and ∆g ∈ L 2 ), then the Markov kernels become Feller in the stronger norm of H 1 × H 1 .
We also study the ergodicity of the electroconvection model (1.5)-(1.7). The existence of an invariant measure for the stochastic NSE system was obtained in [9,16,18], and the ergodic theory for the stochastic NSE became the center of interest of many subsequent papers (cf. [4,11,13,15,34,38,36,11] and references therein). Existence and regularity of invariant measures were obtained in [19] for the three-dimenional stochastic primitive equations. In [6], existence and uniqueness of an ergodic invariant measure was established for the 2D fractionally dissipated periodic stochastic Euler equation.
The dissipative term Λq in (1.5) is critical, and this is a source of technical difficulty. When the potential Φ vanishes, and with a low regular noise process, we use the Krylov Bogoliubov averaging procedure to prove that the stochastic model (1.5)-(1.7) has an invariant measure supported on H 1 2 × H 2 . If the noise processes are smooth then the invariant measures are smooth. This follows from bounds of the form H 2 )dt ≤ Γ 1 (u 0 , q 0 , f, g,g) + Γ 2 (f, g,g)T (1.8) and E T 0 log(1 + q 2 H k+ 1 2 + u 2 H k+2 )dt ≤ Γ 1,k (u 0 , q 0 , f, g,g) + Γ 2,k (f, g,g)T (1.9) for k ≥ 1. These bounds are obtained by taking advantage of the smoothing properties of the Stokes operator and the nonlinear coupling, and employing the logarthmic strategy introduced in [19]. The question of uniqueness of invariant measures requires a deeper structural understanding of the interplay of the dynamics and stochastic perturbation. A number of approaches have been used in the recent literature ( [4,12,27,28,31,32,33,38,36] and references therein). In this paper we use the asymptotic coupling approach introduced in [26] and [28]. The asymptotic coupling framework was used in [21] to obtain uniqueness of the invariant measures of stochastically forced Navier-Stokes equations, fractionally dissipative Euler equations and damped nonlinear wave equations. In order to show that a stochastic differential equation with initial data y(0) = y 0 has only one ergodic measure, the idea is to build a copy dỹ = F (ỹ)dt + G(y,ỹ)1 t≤τ dt + d l=1 σ l dW l (1.11) where the feedback control G is such that y andỹ are forced to approach each other, y(t)−ỹ(t) → 0 in an appropriate norm, on the event {τ = ∞} where τ is a stopping time such that the coupled system (1.10)-(1.11) has global solutions with initial dataỹ(0) =ỹ 0 , and P(τ = ∞) > 0 . Moreover, it is required that ∞ 0 σ −1 G(y(t),ỹ(t)) 2 1 t≤τ dt < C (1. 12) holds (for a.e. w ∈ Ω) for some deterministic constant C > 0. If such a construction can be done, then (1.10) has a unique ergodic invariant measure. Finding an appropriate feedback G is typically based on splitting a Hilbert space X into the direct sum of a finite-dimensional space X low and an infinite-dimensional space X high X = X low ⊕ X high (1.13) in such a way that the long time dynamics are controlled by the low frequency part in X low . More precisely, the property used is that if the low frequency parts of two solutions are asymptotically the same, then the high frequency parts in X high are also asymptotically the same. Accordingly, two realizations of (1.10) are coupled in such a way that that their low frequency parts coincide for large time t > τ provided that they meet at time t = τ .
The uniqueness of the invariant measure of the electroconvection model (1.5)-(1.7) is obtained by constructing an appropriate feedback control and stopping time. The construction requires L 4 bounds for q and H 2 bounds for u, and exponential martingale estimates and the Burkholder-Davis-Gundy inequality. The main difficulty is due to the weaker dissipation of the charge densities, and here we use ideas from [20] to estimate the feedback control. This paper is organized as follows. In section 3, we show that the system (1.5)-(1.7) has a unique global solution provided that the initial charge density has a zero spatial average and is L 4 integrable, the initial velocity is divergence-free and is weakly differentiable, and the noise is sufficiently regular. The proof is based on uniform estimates in Lebesgue spaces which are established in Appendix A. In section 4, we define the semigroup associated with (1.5)-(1.7) and we prove that it is weak Feller. In the absence of potential (Φ = 0), we show in section 5 the existence of an invariant measure for the Markov transition kernels associated with the electroconvection model (1.5)-(1.7) based on the Krylov-Bogoliubov averaging procedure under low regularity assumptions imposed on the noises. In section 6, we prove that any invariant measure of (1.5)-(1.7) is smooth provided that the model is forced by smooth noises. Using asymptotic coupling techniques, we prove in section 7 the uniqueness of the invariant measure. In section 8, we address Feller properties in Sobolev norms when the noise processes are sufficiently regular. This uses uniform bounds for the pathwise solution, and these are presented in Appendix B.

BASIC FUNCTIONAL SPACES AND NOTATIONS
For s > 0, we denote by H s (T 2 ) the Sobolev spaces of measurable periodic functions f from T 2 to R (or R 2 ) obeying For a Banach space (X, ⋅ X ) and p, q ∈ [1, ∞], we consider the Lebesgue Banach spaces L p (Ω; L q loc (0, ∞; X)) of functions f from X to R (or R 2 ) satisfying for any T > 0, with the usual convention when p = ∞ or q = ∞. The spaces L q loc (0, ∞; X) and L p (Ω; C 0 (0, ∞; X)) are defined similarly. Here C 0 (0, ∞; X) is the space of functions f with the property that the map is continuous for any f ∈ X.
For s ∈ R, the fractional Laplacian Λ s applied to a mean zero scalar function f is defined as a Fourier multiplier with symbol k s , that is, for f given by we have that Finally, the periodic Riesz transforms R = (R 1 , R 2 ) applied to scalar functions f are defined as Fourier multipliers and they are bounded operators on L p (T 2 ), 1 < p < ∞. We write R = ∇Λ −1 . Throughout the paper, C denotes a positive universal constant, and C(a, b, c, ...) denotes a positive constant depending on a, b, c, ...

EXISTENCE AND UNIQUENESS OF SOLUTIONS
Let (Ω, F , P ) be a probability space, {F s } s≥0 be a filtration on (Ω, F , P ), and {W k } k≥1 be a collection of independent, identically distributed, real-valued, standard Brownian motions relative to the filtered probability space.
We consider the stochastic electroconvection model on T 2 with initial data q(x, 0) = q 0 and u(x, 0) = u 0 . The unknowns q(x, t, w), u(x, t, w) = (u 1 (x, t, w), u 2 (x, t, w)), and p(x, t, w) depend on three different variables: position x ∈ T 2 , time t ∈ [0, ∞), and outcome w ∈ Ω. The body forces f and the potential Φ depend only on the position variable x. The forces f are smooth, divergence-free and have a zero space average. The potential Φ is assumed to be smooth. The noise termsgdW and gdW are given bỹ We assume that the functionsg l are mean-zero and the vector fields g l are divergence-free for all l ∈ N. For k ≥ 0 and p > 0, we denote and and g ∈ H k ,g ∈ H k , org ∈ L p if the quantities (3.4), (3.5), or (3.6) are finite respectively. In this section, we prove the existence and uniqueness of solutions of the stochastic model (3.1): Theorem 1. Fix a stochastic basis (Ω, F , P, {F t } t≥0 , W ). Let q 0 ∈ L 4 have mean zero over T 2 , and let u 0 ∈ H 1 be divergence-free. Supposeg ∈ L 4 , g ∈ H 1 , f ∈ L 2 , and ∆Φ ∈ L 4 . Then there exists a unique pair (q, u) such that q is mean-free, u is divergence-free, (3.8) Moreover, the elements (q, u) are F t adapted and obey for any ξ ∈ H 1 and a.e. w ∈ Ω, and for any v ∈ H 1 and a.e. w ∈ Ω.
For each ǫ ∈ (0, 1], we let J ǫ be the standard mollifier operator and we consider the viscous approximation with smoothed out initial data q ǫ 0 = J ǫ q 0 , u ǫ 0 = J ǫ u 0 . For each ǫ ∈ (0, 1], the viscous system (3.11) is forced by smooth noise processes and has local smooth solutions, a fact that can be shown using a fixed point iteration technique. These local solutions extend to global smooth solutions as they remain uniformly bounded in all Sobolev norms, a result that follows from energy-type arguments (see for instance Appendix B). In Proposition 1 below, we establish bounds, uniform in time and ǫ, for the solutions of (3.11) in Lebesgue spaces. These estimates are needed to apply the drift identification argument of [37] and prove Theorem 1. Proposition 1. Let q 0 ∈ L 4 have mean zero over T 2 . Let u 0 ∈ H 1 be divergence-free. Supposẽ g ∈ L 4 and g ∈ H 1 . Then the solution (q ǫ , u ǫ ) of (3.11) satisfies for any p ≥ 2, for any p ≥ 2, and (3.15) The proof of Proposition 1 is based on several applications of Itô's lemma and is presented in Appendix A.
Let q 1 ∈ L 4 , q 2 ∈ L 2 , u 1 ∈ H 2 and u 2 ∈ H 1 . Then there is a positive universal constant C 0 such that Integrating by parts, we have (3.20) By Hölder and Young inequalities, we have We note that in view of the divergence-free condition satisfied by u 2 , and hence where we used Ladyzhenskaya's interpolation inequality applied to u 1 − u 2 . Now, we write Adding (3.24) and (3.25), four terms cancel out, namely due to the divergence-free condition satisfied by u 2 − u 1 . We estimate using Hölder's inequality, the boundedness of the Riesz transforms in L 4 , Ladyzhenskaya's inequality, and Young's inequality. In view of the commutator estimate (see [1,Proposition 3]) that holds for any divergence-free v ∈ H 2 and mean-zero ρ ∈ L 2 , we have Here we also used that u 1 is divergence-free. Collecting the bounds (3.20)-(3.30) and applying them to (3.19), we obtain where K(Φ, u 1 , q 1 ) is given by (3.47). This finishes the proof of Proposition 2. Now, we prove Theorem 1.
Putting (3.52) and (3.53) together, we obtain , Ψ 1 having mean zero and Ψ 2 being divergence-free, we obtain (3.56) We divide by λ, and then take the limit as λ goes to zero. We obtain (3.48) from which we conclude that F 0 = F (q, u).
Uniqueness of solutions is obtained as for the deterministic system [1, Theorem 2]. Indeed, if we suppose the existence of two different solutions, and we write the equations obeyed by their difference, then we obtain deterministic equations which are independent of the noise. We omit further details. Remark 1. The existence of unique pathwise solutions can be obtained by setting and

59)
and passing to the limit using the Aubin-Lions lemma. However, this requires higher regularity assumptions on the noise processes forcing the system (as shown in Proposition 17 below). Consequently, the identification of drift technique minimizes the regularity conditions imposed on the noises g andg.
Remark 2. If the ranges ofg and g are infinite countable and their components are time-dependent, then the existence and uniqueness of solutions to the corresponding stochastic electroconvection model are obtained on the time interval [0, T ] provided that the following regularity condition holds.

ELECTROCONVECTION SEMIGROUP AND WEAK FELLER PROPERTIES
We consider the space consisting of vectors (ξ, v) where ξ ∈ H − 1 2 has mean zero and v ∈ L 2 is divergence-free, and we consider the space consisting of vectors (ξ, v) where ξ ∈ L 4 has mean zero and v ∈ H 1 is divergence-free. We define the norms ⋅ H and ⋅ V by and respectively. For a time t ≥ 0 and a Borel set A ∈ B(V), we define the Markov transition kernels associated with (3.1) by where (q, u)(t, (q 0 , u 0 )) denotes the solution of the stochastic model (3.1) with initial data (q 0 , u 0 ) at time t. Let M b (V ) be the collection of bounded real-valued Borel measurable functions on V. For each t ≥ 0 and ϕ ∈ M b (V ), we define the Markovian semigroup (which will also be denoted by {P t } t≥0 ) by We point out that continuity of ϕ on the space has the following weak Feller properties: In the proof of Theorem 2 presented below, we use Propositions 3 and 4.
Proof: We write the equations obeyed by the differences q 1 − q 2 and u 1 − u 2 , and we take their L 2 inner product with Λ −1 (q 1 − q 2 ) and u 1 − u 2 respectively. We add the resulting energy equalities and we obtain 1 2 where F is given by (3.16). In view of (3.17), we have where r(t, q 1 , u 1 ) is given by (3.47). Multiplying by the integrating factor e − ∫ t 0 r(s)ds and integrating in time from 0 to t give (4.8).
Proof: By Itô's lemma, we have and We add the equations (4.13) and (4.14). Integrating by parts, we have and using the cancellation we obtain the differential equation From (4.17), we arrive at the differential inequality Integrating in time from 0 to t, taking the supremum over [0, T ], applying the expectation E in w, and using suitable martingale estimates, we obtain (4.12). This completes the proof of Proposition 4. Now we prove Theorem 2: In view of the continuity property given in Proposition 3, we have and and hence Eϕ((q, u)(t, (ξ n , v n ))) → Eϕ((q, u)(t, (ξ, v))) (4.25) by the Lebesgue Dominated Convergence Theorem, which can be applied due to the growth condition (4.7), the bound (4.12), and the convergence (4.21) yielding the boundedness of the sequence This ends the proof of Theorem 2.

EXISTENCE AND REGULARITY OF INVARIANT MEASURES IN THE ABSENCE OF POTENTIAL
In this section, we consider the electroconvection system We note that the system (5.1) is in the mean-zero frame: if the initial charge density and velocity are assumed to have a zero spatial average, then the solution (q, u) will have mean zero over T 2 for all positive times t ≥ 0. LetL p andḢ s be the spaces of L p and H s functions with zero spatial averages respectively. Let H and V be the spaces of L 2 and H 1 functions that are divergence-free and mean zero respectively.
respectively. We note thatV is compactly embedded inḢ. We define the operator A on where the sequence of eigenvalues {λ k } ∞ k=1 of A counted with multiplicity is nondecreasing and diverges to ∞. Asymptotically, λ k ≥ ck for k ≥ 1. Let P N and Q N be the orthogonal projections oḟ H onto the space spanned by the first N eigenfunctions of A, (e k , w k ) corresponding to eigenvalues λ k , and its orthogonal complement respectively. We have the inequality which holds for all N ≥ 1.
The Markov transition kernels {P t } t≥0 associated with the electroconvection model (5.1), are defined onV and areḢ-Feller as shown in Theorem 2. Here we establish the existence of invariant measures for the Markov transition kernels {P t } t≥0 .
Theorem 3. Suppose that g ∈ V andg ∈L 4 . There exists an invariant measure µ for the Markov transition kernels associated with (5.1). Moreover for any invariant measure µ of (5.1), where C is positive constant depending only on f L 2 , g H 1 , and g L 4 .
The proof of Theorem 3 uses the following auxiliary propositions and is presented at the end of this section. All the estimates can be done rigorously by taking a viscous system approximating (5.1), deriving the bounds for the mollified solution, and then inheriting them to the solution of (5.1) using the lower semi-continuity of the norms. We present formal proofs, omitting the approximation.
holds for all t ≥ 0.
Proof: The sum of the H − 1 2 norm of q and L 2 norm of u obeys the energy equality (cf. (4.13)-(4.17) above) which gives the differential inequality where we used the Poincaré inequality to bound L 2 norm of the mean-free vector u by the L 2 norm of its first order derivative. We integrate in time from 0 to t and we apply E. We obtain the desired bound (5.11).
holds for all t ≥ 0.
Proof: The L 2 norm of q evolves according to where we used the cancellation (u ⋅ ∇q, q) L 2 = 0. We integrate in time from 0 to t and we apply E. We obtain (5.14).
holds for all t ≥ 0.
Proof: The p-th power of the L 4 norm of q obeys the energy inequality Integrating in time from 0 to t and applying E, we obtain the desired bound (5.16).
Proof: The L 2 norm of ∇u obeys Here we used the identity (u ⋅ ∇u, ∆u) L 2 = 0 (5.20) that holds in the two-dimensional periodic setting on T 2 . In view of the boundedness of the Riesz transforms on L 4 , we have Consequently, an application of Young's inequality yields Integrating in time from 0 to t and applying E, we obtain In view of the bound (5.16) applied with p = 4, we obtain (5.18).
Proof: Suppose u 0 = q 0 = 0. Let R > 0, and let B R be the ball of radius R inL 2 × V (which is compact inḢ). By Chebyshev's inequality, as R → ∞ in view of the bound (5.11) that is linear in T . Therefore, the family {ν T } is tight inḢ, ending the proof of Proposition 9. Now we prove Theorem 3. Proof of Theorem 3: We adapt the notation w = (q, u) and write solutions as w(t, w 0 ). From the weak Feller property obtained in Theorem 2, the tightness of the time-averaged measures obtained in Proposition 9, and the Krylov-Bogoliubov averaging procedure, we conclude that there exists a probability measure µ satisfying for any T > 0 and any ϕ ∈ C b (Ḣ). Now we study the regularity of µ and we prove (5.10). For n ≥ 1, we let P n be the projection onto the space spanned by the first n eigenfunctions of −∆. For n ≥ 1, R > 0, w = (q, u) ∈Ḣ, we let Ψ n,R (w) = P n q 2 L 2 + ∇P n u 2 L 2 ∧ R (5.27) and we note that Ψ n,R ∈ C b (Ḣ). In view of (5.11), we estimate for any T > 0. Let BḢ(ρ) be the ball Then, using invariance, we have We choose ρ large enough so that Rµ(Ḣ ∖ BḢ(ρ)) ≤ 1 (5.31) and then we choose T large enough so that and we get and by the Monotone Convergence Theorem, we obtain Therefore, the invariant measure µ is supported on X 2 =L 2 × V . Next we upgrade the regularity of the measure µ. For w = (q, u) ∈ X 2 , we define In view of the bounds (5.11) and (5.14), we have for any T > 0. Letting B X 2 (ρ) be the ball we use (5.37) and invariance to obtain We choose ρ large enough and T large enough so that By Fatou's lemma and the Monotone Convergence Theorem, we obtain Therefore, the invariant measure µ is supported on In view of the bounds (5.14) and (5.18), we have Using the bound (5.37), invariance, and the continuous embedding of H 1 2 in L 4 , we obtain ). (5.45) We choose ρ large enough and T large enough so that Therefore, the invariant measure µ is supported onḢ . This ends the proof of Theorem 3.

HIGHER REGULARITY OF INVARIANT MEASURES
In this section, we prove that any invariant measure of (5.1) is more regular thanḢ Theorem 4. Suppose g andg are smooth. If µ is an invariant measure of (5.1), then µ is smooth and satisfies for any k ≥ 0.
The proof of Theorem 4 is based on the following auxilliary propositions and is presented at the end of this section. Proposition 10. Let u 0 ∈ V and q 0 ∈L 4 . Suppose g ∈ V andg ∈L 4 . Let p ≥ 4. Then holds for all t ≥ 0.
Proof: The L 2 norm of ∇u evolves according to the stochastic energy equality Consequently, the p-th power of ∇u L 2 obeys In view of the Poincaré inequality, we obtain We integrate in time from 0 to t and we apply E. In view of the bound (5.16), we obtain (6.2).
Proof: The stochastic process ∇u 4 The 4-th power of the L 2 norm of ∇u evolves according to whereas the 4-th power of the L 4 norm of q evolves according to d q 4 L 4 = −4(Λq, q 3 ) L 2 dt + 6(g 2 , q 2 ) L 2 dt + 4(g, q 3 ) L 2 dW. (6.9) Consequently, the product ∇u 4 L 2 q 4 L 4 satisfies the energy equality d q 4 which yields the energy inequality Here, we used the nonlinear Poincaré inequality for the fractional Laplacian in L 4 applied to the mean zero function q (see [1,6]) By the Cauchy-Schwartz inequality, Young's inequality and the Poincaré inequality applied to the mean zero function ∇u, we estimate The boundedness of the Riesz transforms on L 4 yields using Young's inequality. Finally, we estimate Putting (6.11)-(6.17) together, we obtain the differential inequality We integrate in time from 0 to t and we apply E. The bound (5.16) applied with p = 4 and p = 12 together with the bound (6.2) gives the desired estimate (6.6).
Proof: TheḢ (6.22) By Itô's lemma, we have The nonlinear term is estimated using commutator estimates (see [1, Proposition 3]) After applying Young's inequality, we obtain Next, we integrate in time from 0 to t, apply E, and obtain Therefore, Xds. (6.28) In view of the bounds (5.14) and (5.18), we obtain (6.19), completing the proof.
Proof: By Itô's lemma, we have In order to estimate the nonlinear term, we integrate by parts, use the divergence-free property ∇ ⋅ u = 0, to obtain and Ladyzhenskaya's interpolation inequality. We obtain Hence, an application of Young's inequality yields We integrate in time from 0 to t and we apply E. In view of (6.2) and (6.6), we obtain (6.29).
holds for all t ≥ 0, then the following estimate holds for all t ≥ 0.
Proof: The Itô lemma yields and 42) and Then the stochastic process X evolves according to An application of Itô's lemma gives the stochastic energy equality from which we obtain the following differential inequality In view of the commutator estimate that holds for any s > 0, p ∈ (1, ∞), p 2 , p 3 ∈ (1, ∞), 1 p = 1 p 1 + 1 p 2 = 1 p 3 + 1 p 4 , and all appropriately smooth functions F and G (see [ Here, we used the continuous Sobolev embedding of H 1 2 in L 4 . In view of the fractional product estimate that holds for any s > 0, p ∈ (1, ∞), p 2 , p 3 ∈ (1, ∞), 1 p = 1 p 1 + 1 p 2 = 1 p 3 + 1 p 4 , and all appropriately smooth functions F and G (see [33,Lemma A.1 Therefore, we obtain the inequality which boils down to after application of Young's inequality. We integrate in time from 0 to t and we apply E. Using the bounds (5.18) and (6.29), and applying Young's inequality, we conclude that E t 0X 1 + X ds ≤ log(1 + X(0)) + C(f, g,g, k)( ∇q 0 12 L 2 + ∇u 0 for all t ≥ 0. Bounding similarly to (6.28), we have E t 0 log(1 +X)ds ≤ log(1 + X(0)) + C(f, g,g, k)( ∇q 0 12 L 2 + ∇u 0 In view of (6.36), we obtain (6.37). We end this section by proving Theorem 4.

Proof of Theorem 4:
Suppose µ is an invariant measure of (5.1). By Theorem 3, µ is supported on H 1 2 × (H 2 ∩ H). In view of the bounds (6.19) and (6.29), and repeating the same argument used to prove Theorem 3, we conclude that µ is supported on H 3 2 × (H 2 ∩ H). Now we bootstrap using Proposition 14 and we deduce that µ is supported on H k+ 3 2 × H k+3 for any k ≥ 0. This shows that µ is smooth and completes the proof of Theorem 4.

UNIQUENESS OF INVARIANT MEASURES
In this section, we prove that (5.1) has a unique ergodic invariant measure provided that the ranges ofg and g are large enough in phase space. Uniqueness is obtained by employing asymptotic coupling arguments from [21].
Theorem 5. Suppose that g ∈ V andg ∈L 4 . There exists N = N(f, g,g) such that if P NḢ ⊂ range(g, g), then (5.1) has a unique ergodic invariant measure.
In order to prove Theorem 5, we need the following proposition: Proposition 15. Let R > 0. Then there exist positive universal constants c and C such that the estimates hold.
Proof: We integrate in time from 0 to t the differential inequality (see (5.19)) and take the supremum over t ≥ 0 to obtain Exponential martingale inequalities [21, (3.4)] imply (7.5) Therefore (7.1) is established. The derivation of (7.2) is based on ideas from [20]. Indeed, the L 4 norm of q evolves according to d q 4 L 4 + 4(Λq, q 3 ) L 2 dt = 6(g 2 , q 2 ) L 2 dt + 4(g, q 3 ) L 2 dW. (7.6) (see (5.17)). By the Poincaré inequality for the fractional Laplacian in L 4 , we have Thus, we obtain the differential inequality We integrate from 0 to t, and take the supremum over t ≥ 0. We obtain We have M(s) . (7.14) Using the Burkholder-Davis-Gundy inequality [30] EM we obtain Here we used the estimate (5.16) applied for p = 12. Therefore, in view of the Chebyshev's inequality. This gives (7.2) ending the proof of Proposition 15. Finally, we prove the uniqueness result: Proof of Theorem 5: Fix (q 0 , u 0 ) and (Q 0 , U 0 ) inV. Our aim is to establish the conditions for the asymptotic coupling framework presented in Section 2.4 of [21]. To this end, we consider (q, u) solving (5.1) with (q(0), u(0)) = (q 0 , u 0 ), and (Q, U) solving and K, N and λ are positive constants to be determined later. By Girsanov's theorem [21, Theorem 2.2], the law of (Q, U) is absolutely continuous with respect to the solution (q, u)(⋅, (Q 0 , U 0 )) of (5.1) corresponding to (Q 0 , U 0 ) for any choices of λ > 0 and K > 0. Consequently, the uniqueness of the invariant measure follows from an application of Corollary 2.1 in [21], provided that we can find some positive constants λ and K such that (q, u) − (Q, u) → 0 in the norm ofḢ on a set of positive measure.
Let ω = (ξ, v). Taking the L 2 inner product of (7.22) with (Λ −1 ξ, v), we obtain the differential inequality 1 2 where we used the cancellations (U ⋅ ∇v, v) L 2 = 0 (7.24) and (v ⋅ ∇Q, and using Hölder's inequality, Ladyzhenskaya's interpolation inequality, Young's inequality, the boundedness of the Riesz transform on L 4 , and the commutator estimate (3.29). This yields the differential inequality For a fixed integer N, we have N +1 in view of the inequality (5.8). Hence Integrating in time, we obtain for any t ∈ [0, τ K ]. For R ≥ 0, we consider the sets and By Proposition 15, we have P(E R ∩ F R ) > 0 when R is sufficiently large. Indeed, when R is large. Consequently, on E R ∩ F R and for t ∈ [0, τ K ], we have N+1 +C(f,g,g) t e C( ∇u 0 L 2 , q 0 L 4 ,R) . (7.36) We choose an integer N = N(f, g,g) large enough so that yielding inḢ as t → ∞. This completes the proof of Theorem 5.

FELLER PROPERTY IN THE H 1 NORM
We consider the spaceṼ In this section, we show that the transition kernels associated with (5.1) are Feller in the norm ofṼ.
We need the following propositions.
Proof: Let q = q 1 − q 2 and u = u 1 − u 2 . The norm ∇q L 2 satisfies the energy inequality where we integrated by parts and used the divergence-free condition of u 2 and u. Applying Young's inequality and using the continuous embedding of H 1 2 in L 4 , we obtain d dt On other hand, the norm ∇u L 2 obeys Adding (8.6) and (8.8), we get d dt which gives (8.3).
Proposition 17. Suppose ∇g ∈ L 8 and ∆g ∈ L 2 . Let (q 0 , u 0 ) ∈V and T > 0. Then the solution (q, u) to the system (5.1) is uniformly bounded (almost surely) in by some constant depending only on g,g, f, ∇u 0 L 2 and q 0 almost surely.

APPENDIX A. UNIFORM BOUNDS IN LEBESGUE SPACES
In this Appendix, we prove Proposition 1. For simplicity, we ignore the viscous term −ǫ∆q ǫ in (3.1) because it does not have any major contribution in estimating the solutions of the mollified system (3.11) and vanishes as we take the limit ǫ → 0. We also drop the ǫ superscript.
The proof is divided into 7 main steps.
Step 1. We prove that the estimate (3.12) holds when p = 2. Proof of Step 1. By Itô's lemma, we have We integrate in the space variable over T 2 . In view of the divergence-free condition obeyed by u, the nonlinear term vanishes, that is which yields the energy equality using the Hölder and Young inequalities. We obtain the differential inequality Integrating in time from 0 to t, we get We take the supremum over all t ∈ [0, T ], Now we apply the expectation E. In view of the martingale estimate (see [9]), This gives (3.12) when p = 2.
Step 2. We prove that the estimate (3.12) holds for any p ∈ [4, ∞). Proof of Step 2. Applying Itô's lemma to the process F (X t (w)) where X t (w) = q(t, w) 2 L 2 obeys (A.3) and F (ξ) = ξ p 2 , we derive the energy equality which yields the differential inequality In view of the bound where we used Young's inequality to estimate and Integrating in time (A.13) from 0 to t and taking the supremum over [0, T ], we obtain and we obtain (3.12).
Step 3. We show that the velocity u obeys Proof of Step 3. We apply Itô's lemma pointwise in x and we obtain the energy equality where we used the cancellation (u ⋅ ∇u, u) L 2 = 0 (A. 21) due to the divergence-free condition satisfied by u. By Ladyzhenskaya's interpolation inequality .22) and the boundedness of the Riesz transforms in L 4 , we estimate We also estimate using Hölder's inequality followed by Young's inequality. We obtain the differential inequality for all s ∈ [0, t]. Integrating in time from 0 to t, we obtain (A.28) We take the supremum in time over [0, T ] and apply E. Using the continuous Sobolev embedding and (3.12) with p = 4, we have and we obtain (A.18).
Step 4. We prove that (3.13) holds for p = 4. Proof of Step 4. By Itô's lemma, we have Integrating in the space over T 2 , we obtain the energy equality We note that (u ⋅ ∇q, q 3 ) L 2 = 0 (A.35) due to the divergence-free condition for u. By the nonlinear Poincaré inequality for the fractional Laplacian in L 4 applied to the mean zero function q, we have Using Hölder's inequality with exponents 4, 4 3 and Young's inequality with exponents 4, 4 3, we get , (A.38) using Hölder and Young inequalities. Putting (A.34)-(A.38) together, we obtain the differential inequality for all t ∈ [0, T ]. We take the supremum over [0, T ] and then we apply E. We estimate and we obtain (3.13) for p = 4.
Proof of Step 5. The stochastic energy equality holds for any p ≥ 8. By Hölder's inequality with exponents 4 3, 4 and Young's inequality with exponents p (p − 2), p 2, we have We obtain Integrating (A.44) in time from 0 to t, taking the supremum over [0, T ], applying E, and estimating E sup we obtain (3.13).
Step 6. We show that (3.14) holds. Proof of Step 6. We derive the stochastic energy equality By Young's inequality with exponents p (p − 2) and p 2, and Similarly, using Young's inequality with exponents p (p − 1) and p, By Ladyzhenskaya's interpolation inequality and the boundedness of the Riesz transforms in L 4 (T 2 ), we have This yields the differential inequality and thus d e −t u p L 2 (s) + e −s u p−2 L 2 ∇u 2 L 2 ds ≤ e −s C(p) g p L 2 ds + C(p) f p L 2 ds + C(p) ∇Φ p L ∞ q p L 2 ds + C(p) q 2p L 2 ds + C(p) q 2p L 4 ds + pe −s u p−2 L 2 (g, u) L 2 dW. (A.53) We integrate in time from 0 to t, take the supremum over [0, T ], and apply E. We obtain Putting (A.54) and (A.55) together, and using (3.12) and (3.13), we obtain (3.14).
Step 7. We prove that (3.15) holds. Proof of Step 7. We write the equation satisfied by ∇u, apply Itô's lemma, and integrate in the space variable. We obtain the energy equality The nonlinear term for the velocity vanishes, that is (u ⋅ ∇u, ∆u) L 2 = 0, (A.57) and using Hölder's inequality, we obtain An application of Young's inequality yields the differential inequality We integrate (A.59) in time from 0 to t, take the supremum in time, and then apply E. We obtain We estimate the martingale term E sup Putting (A.60) and (A.61) together, and using (3.12) with p = 2 and (3.13) with p = 4, we get (3.15).

APPENDIX B. PATHWISE UNIFORM BOUNDS FOR THE SOLUTIONS
In this section, we prove Proposition 17. We let (q, u) be the solution to (5.1) corresponding to the initial data (q 0 , u 0 ). Letφ where we used the divergence-free condition imposed on g.
Step 1. Bounds for the velocity in L 2 loc (0, ∞; H 1 (T 2 )). We take the L 2 inner product of the Q equation with Q, and we obtain We estimate the nonlinear term using Hölder's inequality, Ladyzhenskaya's interpolation inequality applied to the mean zero function U, the continuous Sobolev embedding H 1 2 ⊂ L 4 , and Young's inequality. This yields the differential inequality Now we take the L 2 inner product of the Q equation with Λ −1 Q and we get 1 2 Integrating by parts and using the divergence-free condition obeyed by U + φ, we can rewrite the nonlinear term as − and we estimate where we have used the boundedness of the Riesz transforms on L p (T 2 ) for p ∈ (1, ∞). We obtain Finally, we take the L 2 inner product of the equation obeyed by U with U and we obtain We integrate by parts the nonlinear term. Using the fact that U + φ is divergence-free, we have This yields the differential inequality 1 2 We add (B.8), (B.12) and (B.15). Setting where A(t) and B(t) are some positive constants depending on φ,φ and f . This implies for all t ≥ 0.
Step 2. Bounds for the charge density in L ∞ loc (0, ∞; L 4 (T 2 )). We take the L 2 inner product of the Q equation with (Q) 3 . Using the Poincaré inequality for the fractional Laplacian, we get the deterministic differential inequality 1 4 In view of the continuous Sobolev embedding of H 1 (T 2 ) in L 8 (T 2 ), we bound the nonlinear term Integrating in time from 0 to t and using the boundedness of ∇U in L 2 loc (0, ∞; L 2 (T 2 )) derived in Step 1, we obtain uniform in bounds for the L 4 norm of Q.
Step 3. Bounds for the velocity in L 2 loc (0, ∞; H 2 (T 2 )). Taking the L 2 inner product of the equation obeyed by U with −∆U, we get Since the trace of M T M 2 vanishes for any two-by-two traceless matrix M, we have We obtain d dt We integrate in time from 0 to t and we use the bounds derived in Step 1 and Step 2 to obtain uniform bounds for ∇U L 2 and ∫ t 0 ∆U 2 L 2 ds.