Sandra Cerrai

Arnaud Debussche

Large deviations for the two-dimensional stochastic Navier-Stokes equation with vanishing noise correlation *

We are dealing with the validity of a large deviation principle for the two-dimensional Navier-Stokes equation, with periodic boundary conditions, perturbed by a Gaussian random forcing. We are here interested in the regime where both the strength of the noise and its correlation are vanishing, on a length scale and δ( ), respectively, with 0 < , δ( ) << 1. Depending on the relationship between and δ( ) we will prove the validity of the large deviation principle in different functional spaces.

Introduction

We are dealing here with the following randomly forced two-dimensional incompressible Navier-Stokes equation with periodic boundary conditions, defined on the domain D = [0, 2π] 2 ,    ∂ t u(t, x) = ∆u(t, x) -(u(t, x) • ∇)u(t, x) + ∇p(t, x) + √ ∂ t ξ δ (t, x), x ∈ D, t ≥ 0, div u(t, x) = 0, x ∈ D, t ≥ 0, u(0, x) = u 0 (x), x ∈ D.

(1.1) Here u denotes the velocity and p denotes the pressure of the fluid. Moreover, ξ δ (t, x) denotes a Gaussian random forcing. We are interested in the regime where the noise is weak, that is its typical strength is of order √ << 1, and almost white in space, that is its correlation decays on a length-scale δ << 1.

As well known, in order to have well posedness in C([0, T ]; [L 2 (D)] 2 ) for equation (1.1), the Gaussian noise ξ δ cannot be white in space. In fact, white noise in space and time has been considered in [START_REF] Da Prato | Two-dimensional Navier-Stokes equations driven by a spacetime white noise[END_REF], where the well-posedness of equation (1.1) has been studied in suitable Besov spaces of negative exponent, for µ -almost every initial condition, where µ is a suitable centered Gaussian measure, depending on > 0. It turns out that, for different values of > 0, the measures µ are all singular, so that the result proved in [START_REF] Da Prato | Two-dimensional Navier-Stokes equations driven by a spacetime white noise[END_REF] does not imply the well posedness of equation (1.1) for any initial datum in some subset of the Besov space that remains independent of > 0.

In the present paper, we assume that for any fixed δ > 0 the noise ξ δ (t, x) is sufficiently smooth in the space variable x ∈ D to guarantee that for any initial condition u 0 ∈ [L 2 (D)] 2 there exists a unique generalized solution in C([0, T ]; [L 2 (D)] 2 ) (see Section 2 for all details). As a consequence of the contraction principle and of some continuity properties of the solution of equation (1.1), for any δ > 0 fixed, the family {L(u ,δ )} >0 , given by the solutions of equation (1.1), satisfies a large deviation principle in C([0, T ]; [L 2 (D)] 2 ), for any T > 0 fixed, with rate and action functional

I δ T (f ) = 1 2 T 0 |Q -1 δ f (t) -Af (t) -b(f (t)) | 2 [L 2 (D)] 2 dt,
where A is the Stokes operator, b is the Navier-Stokes nonlinearity and Q δ is the square root of the covariance of the noise ξ δ (see Section 2 for all definitions and notations and also [START_REF] Brzeźniak | Large deviations principle for the invariant measures of the 2D stochastic Navier-Stokes equations on a torus[END_REF]).

In [START_REF] Brzeźniak | Quasipotential and exit time for 2D Stochastic Navier-Stokes equations driven by space time white noise[END_REF], the limiting behaviors, as δ ↓ 0, for the large deviation action functional I δ T , as well as for the corresponding quasipotential V δ have been studied. Namely it has been proven that if the operator Q δ converges strongly to the identity operator, and a few other conditions are satisfied, then the operators I δ T and V δ converge pointwise, as δ ↓ 0, to the operator

I T (f ) = 1 2 T 0 |f (t) -Af (t) -b(f (t))| 2 [L 2 (D)] 2 dt, (1.2)
and the operator V (x) = |x| 2 [H 1 (D)] 2 , respectively. Notice that I T and V would be the natural candidates for the large deviation action functional in C([0, T ]; [L 2 (D)] 2 ) and the quasi-potential in [L 2 (D)] 2 , in case equation (1.1), perturbed by space-time white noise, were well-posed in [L 2 (D)] 2 .

In [START_REF] Brzeźniak | Quasipotential and exit time for 2D Stochastic Navier-Stokes equations driven by space time white noise[END_REF] we have first taken the limit in and then in δ. In the present paper we describe what happens in the relevant case the parameter δ is a function of the parameter that describes the intensity of the noise, and lim →0 δ( ) = 0.

(1.3)

Namely, we show that in this case the family {u ,δ( ) } >0 satisfies a large deviation principle in the space C([0, T ]; B σ p (D)), where B σ p (D) is a suitable Besov space of functions, with σ < 0 and p ≥ 2. Moreover, in the case condition (1.3) is supplemented with the condition lim →0 δ( ) -η = 0, (1.4) for some η > 0, we prove that the family {u ,δ( ) } >0 satisfies a large deviation principle in the space C([0, T ]; [L 2 (D)] 2 ), where equation (1.1), corresponding to δ = 0, is ill-posed. In both cases, the action functional that describes the large deviation principle is the operator I T defined in (1.2). We would like to mention the fact that in [START_REF] Hairer | Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions[END_REF] Hairer and Weber have studied a similar problem for the stochastic reaction-diffusion equation

   ∂ t u(t, x) = ∆u(t, x) + c u(t, ξ) -u 3 (t, ξ) + √ ∂ t ξ δ( ) (t, x), x ∈ D, t ≥ 0, u(0, x) = u 0 (x), x ∈ D, (1.5)
where D is a bounded smooth domain, either in R 2 or in R 3 . By using the recently developed theory of regularity structures, they study the validity of a large deviation principle for the solutions {u ,δ( ) } >0 of equation (1.5), in the case condition (1.3) is satisfied. Actually, they prove that if, in addition to (1.3), the following conditions hold lim

→0 log δ( ) -1 = λ ∈ [0, ∞), for d = 2, lim →0 δ( ) -1 = λ ∈ [0, ∞), for d = 3,
then the family {u ,δ( ) } >0 satisfies a large deviation principle in C([0, T ], C η (D)), where C η (D) is some space of functions of negative regularity in space, with respect to the action functional

I λ T (f ) = 1 2 T 0 |∂ t f -∆f + c λ f + f 3 | 2 [L 2 (D)] 2 dt,
for some explicitly given constant c λ , depending on λ and d and such that c 0 = -c. Moreover, they also consider the renormalized equation

     ∂ t u(t, x) = ∆u(t, x) + (c + 3 c (1) δ( ) -9 2 c (2) δ( ) ) u(t, ξ) -u 3 (t, ξ) + √ ∂ t ξ δ( ) (t, x), u(0, x) = u 0 (x), x ∈ D,
where c

δ( ) and c

δ( ) are the constants, depending on the dimension of the underlying space, arising from the renormalization procedure. They prove that, in this case, if (1.3) holds, then the family of solutions {u ,δ( ) } >0 satisfies a large deviation principle in C([0, T ], C η (D)), with action functional I 0 T . Unlike Hairer and Weber, that use techniques from the theory of regularity structures to prove the validity of the large deviation principle, in this paper we use the weak convergence approach to large deviations, as developed in [START_REF] Budhiraja | Large deviations for infinite dimensional stochastic dynamical systems[END_REF] for SPDEs (see also [START_REF] Sritharan | Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise[END_REF], [START_REF] Chueshov | Stochastic 2D hydrodynamical type systems: well posedness and large deviations[END_REF] and [START_REF] Bessaih | Large deviations and the zero viscosity limit for 2D stochastic Navier-Stokes equations with free boundary[END_REF] for some relevant applications of this method). The argument is simpler and gives a stronger result. In particular, we are able to prove that, when condition (1.4) is satisfied, then the family {u ,δ( ) } >0 satisfies a large deviation principle in the space of continuous trajectories with values in the space H itself and not in a functional space of negative regularity. Notice that in [START_REF] Cerrai | Large deviations for the dynamic Φ 2n d model[END_REF] we have studied an analogous problem for the Φ 2n d -model To this purpose, let {ϕ } >0 be any sequence of {F t } t≥0 -predictable processes, taking values in a ball of L 2 (0, T ; [L 2 (D)] 2 ), P-almost surely, such that lim

→0 ϕ = ϕ weakly in L 2 (0, T ; [L 2 (D)] 2 ), P -a.s.
for some {F t } t≥0 -predictable process ϕ taking values in the same ball of L 2 (0, T ; [L 2 (D)] 2 ). As we will explain in Section 3, in order to use the weak convergence approach to large deviations, we have to show that if u ϕ is the solution of the equation

du(t) = [Au(t) + b(u(t)) + Q ϕ (t)] dt + √ dw δ( ) (t), t ≥ 0, u(0) = u 0 , then we have lim →0 |u ϕ -u ϕ | E = 0 P -a.s. (1.6)
where u ϕ the solution of the random problem

du dt (t) = Au(t) + b(u(t)) + ϕ(t) t ≥ 0, u(0) = u 0 ,
and E coincides with the space C([0, T ]; [L 2 (D)] 2 ) or C([0, T ]; B σ p (D)), depending on whether condition (1.4) is satisfied or not. We would like to stress the fact that the proof of (1.6) is quite different, in the two different cases.

If (1.4) is satisfied, then we can work in a Hilbertian framework. For every α > 0 and > 0, we use the splitting

u ϕ = v α,ϕ + z α + Φ ϕ ,
where

Φ ϕ (t) = t 0 e (t-s)A Q δ( ) ϕ (s) ds, t ≥ 0,
and

z α (t) = √ t -∞ e (t-s)(A-α) d wδ( ) (s), t ≥ 0, (1.7) so that u ϕ (t) -u ϕ (t) = [v α,ϕ (t) -v α,ϕ 0 (t)] + z α (t) + [Φ ϕ (t) -Φ ϕ 0 (t)] .
Our aim is to show that there exists a random family {α } >0 such that the three terms on the right hand side above, corresponding to α = α , converge to zero in

L p (Ω; C([0, T ]; [L 2 (D)] 2 )).
In order to prove that we proceed with suitable energy estimates. Here, the key point is the fact that for every p ≥ 1 there exist θ > 0 and a random variable K (p) such that

|z α | C([0,T ];L p (O)) ≤ (α ∨ 1) -θ c p (T ) K (p), P -a.s.
and for any η small enough and any p, q ≥ 1 there exist c 1,η (p, q) and c 2,η (p, q) such that

E |K (p)| q ≤ c 1,η (p, q) δ( ) -η c 2,η (p,q) .
If (1.4) is not satisfied, we have to work with the mild formulation of the equation in the space E T := C([0, T ]; B σ p (D)) ∩ L β (0, T ; B α p (D)), where B σ p (D) and B α p (D) are suitable Besov spaces, with σ < 0 < α, β ≥ 1 and p ≥ 2 satisfying suitable conditions. Also in this case we proceed with a suitable splitting of the solution u ϕ , but we cannot proceed with energy estimates. We consider the decomposition

u ϕ -u ϕ = [v (t) -u ϕ (t)] + z (t),
where z (t) is the process defined in (1.7), corresponding to α = 0. In this case, one of the key points in order to prove (1.6) is showing that

• 0 e (•-s)A b(z (s)) ds E T ≤ c 2 (t) z ⊗ z -ϑ δ( ) I L ρ (0,T ;[H -ρ (D)] 4 ) ,
for a suitable ρ > 1 and a suitable constant ϑ δ( ) such that lim

→0 E |z ⊗ z -ϑ δ( ) I| κ L p (0,T ;[H σ (D)] 4 ) = 0,
for any κ, p ≥ 1 and σ < 0. This follows from arguments analogous to those used in [START_REF] Da Prato | Two-dimensional Navier-Stokes equations driven by a spacetime white noise[END_REF] 2 Notations and preliminaries

We consider here the following incompressible Navier-Stokes equation with periodic boundary conditions on the two-dimensional domain

D = [0, 2π] 2 ,            ∂ t u(t, x) = ∆u(t, x) -(u(t, x) • ∇)u(t, x) + ∇p(t, x) + √ ∂ t ξ δ (t, x), x ∈ D, t ≥ 0, div u(t, x) = 0, x ∈ D, t ≥ 0, u(0, x) = u 0 (x), x ∈ D, u(t, x 1 , 0) = u(t, x 1 , 2π), u(t, 0, x 2 ) = u(t, 2π, x 2 ), (x 1 , x 2 ) ∈ [0, 2π] 2 , t ≥ 0, (2.1
) where 0 < , δ << 1 are some small positive constants. Here ξ δ (t, x) is a Wiener process on [L 2 (D)] 2 , with covariance Q δ to be defined below.

We assume that the initial condition u 0 and the noise ξ δ have zero average in space, so that u(t) remains with zero average for all time. It is not difficult to get rid of this assumption.

In what follows, we will denote by H the subspace of [L 2 (D)] 2 consisting of periodic, divergence free and zero average functions, that is

H = u ∈ [L 2 (D)] 2 : D u(x) dx = 0, div u = 0, u is periodic in D .
H turns out to be a Hilbert space, endowed with the standard scalar product •, • H inherited from [L 2 (D)] 2 . Moreover, we will denote by P the Leray-Helmholtz projection of

[L 2 (D)] 2 onto H. Now, for any k = (k 1 , k 2 ) ∈ Z 2 0 = Z 2 \ {(0, 0)} we define e k (x) = 1 2π k ⊥ |k| e i x•k = 1 2π k ⊥ |k| e i (x 1 k 1 +x 2 k 2 ) , x = (x 1 , x 2 ) ∈ D, k ∈ Z 0 ,
where

k ⊥ = (k 2 , -k 1 ), |k| = k 2 1 + k 2 2 .
It turns out that the family {e k } k∈ Z 2 0 is a complete orthonormal system in H C , the complexification of the space H. For every s ∈ R, we define

H s (D) :=    u : D → R : |u| 2 H s (D) := k∈ Z 2 0 | u, e k | 2 |k| 2s < ∞    .
Next, for q ∈ N, we set δ q := Π 2 q -Π 2 q-1 , where Π n denote the projection of H into H n := span{e k } |k|≤n . Namely

δ q u = 2 q-1 <|k|≤2 q u, e k H e k , u ∈ s∈ R H s (D).
For any σ ∈ R and p ≥ 1, we define

B σ p (D) :=    u ∈ s∈ R H s (D) : q∈ N 2 pqσ |δ q u| p L p (D) < ∞    .
B σ p (D) turns out to be a Banach space, endowed with the norm

|u| B σ p (D) :=   q∈ N 2 pqσ |δ q u| p L p (D)   1 p
. Now, we define the Stokes operator

Au = P ∆u, u ∈ D(A) = H ∩ [H 2 (D)] 2 ,
where P is the Helmodtz projection. It is immediate to check that for any

k ∈ Z 2 0 Ae k = -|k| 2 e k , k ∈ Z 2 0 .
For any r ∈ R, we denote by (-A) r the r-th fractional power of -A, defined on its domain D((-A) r ). It is well known that D((-A) r ) is the closure of the space spanned by {e k } k∈ Z 2 0 with respect to the norm in [H 2r (D)] 2 and the mapping

u ∈ D((-A) r ) → |(-A) r u| H ∈ [0, +∞), defines a norm on D((-A) r ), equivalent to the usual norm in [H 2r (D)] 2
. Moreover, we have that the Leray-Helmholtz projection P maps [H 2r (D)] 2 into D((-A) r ), for every r ∈ R.

Due to the incompressibility condition, the nonlinearity in equation (2.1) can be rewritten as

(u • ∇)v = div (u ⊗ v),
where

u ⊗ v = u 1 v 1 u 1 v 2 u 2 v 1 u 2 v 2 .
In what follows, we shall set

b(u, v) = -P div (u ⊗ v), b(u) = -P div (u ⊗ u). (2.2)
We recall here that, whenever the quantities on the left-hand sides make sense, it holds

b(u), u H = 0, b(u), Au H = 0, (2.3)
(for a proof see e.g. [START_REF] Temam | Navier-Stokes equations and nonlinear functional analysis[END_REF]). Finally, concerning the noisy perturbation ξ δ (t, x) in equation (2.1), it is a Wiener process on [L 2 (D)] 2 and has zero average. In what follows, we shall set

w δ (t) := P ξ δ (t), t ≥ 0.
w δ (t) is now a Wiener process on H, and we assume it can be written as

w δ (t, x) = k∈ Z 2 0 λ k (δ)e k (x)β k (t), t ≥ 0, x ∈ D,
where {e k } k∈ Z 2 0 is the orthonormal basis that diagonalizes the operator A, {β k (t)} k∈ Z 2 0 is a sequence of independent Brownian motions defined on the stochastic basic (Ω, F, {F t } t≥0 , P), and for any δ > 0

λ k (δ) = 1 + δ |k| 2γ -1 2 , k ∈ Z 2 0
, for some fixed γ > 0. In other words, w δ is a Wiener process on H with covariance Q δ = (I + δ(-A) γ ) -1 . We would like to stress that our result easily generalizes to more general covariance operators.

As we mentioned above, in the present paper we are interested in the asymptotic behavior of equation (2.1), as both and δ go to zero. In particular, we shall assume that δ is a function of , such that lim

→0 δ( ) = 0.
In what follows we shall denote by Q the bounded linear operator in H defined by

Q e k = λ k (δ( )) e k , k ∈ N.
Now, if we project equation (2.1) on H, with the notations we have just introduced, it can be rewritten as

du(t) = [Au(t) + b(u(t))] dt + √ dw δ( ) (t), t ≥ 0, u(0) = u 0 . (2.4)
As proven e.g. in [START_REF] Flandoli | Dissipativity and invariant measures for stochastic Navier-Stokes equations[END_REF], equation (2.4) admits a unique generalized solution u ∈ C([0, T ]; H). This means that u is a progressively measurable process taking values in C([0, T ]; H), such that P-a.s. equation (2.4) is satisfied in the integral form

u (t), ϕ H = u 0 , ϕ H + t 0 u (s), Aϕ H ds + t 0 b(u (s), ϕ), u (s) H ds + √ w δ( ) (t), ϕ H , for every t ∈ [0, T ] and ϕ ∈ D(A).
In what follows, for every α ≥ 0 and > 0, we consider the auxiliary Ornstein-Uhlenbeck problem

dz(t) = (A -α)z(t) dt + √ dw δ( ) (t), t ≥ 0, (2.5)
whose unique stationary solution is given by

z α (t) = √ t -∞ e (t-s)(A-α) d w δ( ) (s), t ∈ R. (2.6)
Notice that here w δ( ) (t) is a two sided cylindrical Wiener process, defined by

w δ( ) (t, x) = k∈ Z 2 0 λ k (δ( ))e k (x) βk (t), (t, x) ∈ R × D,
where

βk (t) =    β k (t), if t ≥ 0, βk (-t), if t < 0,
for some sequence of independent Brownian motions { βk (t)} k∈ Z 2 0 , defined on the stochastic basis (Ω, F, {F t } t≥0 , P) and independent of the sequence {β k (t)} k∈ Z 2 0 . It is well known that for any fixed > 0 the process z α belongs to L p (Ω; C([0, T ]; D((-A) β )), for any T > 0, p ≥ 1 and β < γ/2. In the case α = 0, we shall set z (t) := z 0 (t).

(2.7)

3 The problem and the method

We are here interested in the study of the validity of a large deviation principle, as ↓ 0, for the family {L(u )} ∈ (0,1) , where u is the solution of the equation

du(t) = [Au(t) + b(u(t))] dt + √ dw δ( ) (t), t ≥ 0, u(0) = u 0 . (3.1)
Here and in what follows, T > 0 is fixed and

> 0 → δ( ) > 0 is a function such that lim →0 δ( ) = 0. (3.2)
We will prove that depending on the scaling we assume between and δ( ), the family {L(u )} ∈ (0,1) satisfies a large deviation principle in E, where E is a suitable space of trajectories on [0, T ], taking values in some space of functions defined on the domain D and containing H. Then, for any u 0 ∈ H, the family {L(u )} >0 satisfies a large deviation principle in C([0, T ]; H), with action functional

I T (f ) = 1 2 T 0 |f (t) -Af (t) -b(f (t))| 2 H dt. (3.4)
Theorem 3.2. Let → δ( ) be a function satisfying (3.2). Moreover, let σ < 0 and p ≥ 2 be such that

σ > - 2 p ∨ 2 p -1 .
Then, for any u 0 ∈ H θ (D), with θ ≥ σ + 1 -2/p, the family {L(u )} >0 satisfies a large deviation principle in C([0, T ]; B σ p (D)), with the same action function I T introduced in (3.4).

In order to prove Theorems 3.1 and 3.2, we follow the weak convergence approach, as developed in [START_REF] Budhiraja | Large deviations for infinite dimensional stochastic dynamical systems[END_REF]. To this purpose, we need to introduce some notations. For any T > 0, we denote by P T the set of predictable processes in L 2 (Ω × [0, T ]; H), and for any γ > 0, we define the sets

S γ T = f ∈ L 2 (0, T ; H) : T 0 |f (t)| 2 H dt ≤ γ ,

and

A γ T = u ∈ P T : u ∈ S γ T , P -a.s. . Next, for any predictable process ϕ(t) taking values in L 2 ([0, T ]; H), we denote by u ϕ the solution of the problem

du dt (t) = Au(t) + b(u(t)) + ϕ(t) t ≥ 0, u(0) = u 0 . (3.5)
Moreover, for every > 0 we denote by u ϕ (t) the generalized solution of the problem

du(t) = [Au(t) + b(u(t)) + Q ϕ(t)] dt + √ dw δ( ) (t), t ≥ 0, u(0) = u 0 . (3.6) Notice that w δ( ) (t) = Q δ( ) w(t)
, where

w(t, x) = ∞ k=1 e k (x)β k (t).
Then, by using the notations introduced in [5], we have

u ϕ = G √ w + • 0 ϕ(t) dt ,
where G (ψ) denotes the solution f of the problem

df (t) = [Af (t) + b(f (t))] dt + Q δ( ) dψ(t), f (0) = u 0 .
As for equation (2.4), for any fixed ≥ 0 and for any T > 0 and κ ≥ 1, equation (3.6) admits a unique generalized solution u ϕ in L κ (Ω; C([0, T ]; H)). As a particular case ( = 0), we have also well-posedness for equation (3.5).

By proceeding as in [START_REF] Budhiraja | Large deviations for infinite dimensional stochastic dynamical systems[END_REF], the following result can be proven. Theorem 3.3. Let E be a Polish space of trajectories defined on [0, T ] with values in a space of functions defined on the domain D and containing the space H, and let I T be the functional defined in (3.4). Assume that 1. the level sets {I T (f ) ≤ r} are compact in E, for every r ≥ 0;

for every family {ϕ } >0 ⊂ A γ

T that converges in distribution, as ↓ 0, to some ϕ ∈ A γ T , in the space L 2 (0, T ; H), endowed with the weak topology, the family {u ϕ } >0 converges in distribution to u ϕ , as ↓ 0, in E.

Then the family {L(u )} >0 satisfies a large deviation principle in E, with action functional I T .

Actually, as shown in [START_REF] Budhiraja | Large deviations for infinite dimensional stochastic dynamical systems[END_REF], the convergence of u ϕ to u ϕ implies the validity of the Laplace principle in E with rate functional I T . This means that, for any continuous mapping Γ :

E → R it holds lim →0 -log E exp - 1 Γ(u ) = inf f ∈ E ( Γ(f ) + I T (f ) ) . (3.7)
And, once one has shown that the level sets of I T are compact in E, the validity of the Laplace principle as in (3.7) is equivalent to say that the family {L(u )} >0 satisfies a large deviation principle in E, with action functional I T . The proof of condition 1 in Theorem 3.3 is obtained once we show that, when the space L 2 (0, T ; H) is endowed with the topology of weak convergence, the mapping

ϕ ∈ L 2 (0, T ; H) → u ϕ ∈ E,
is continuous. More precisely, condition 1 will follow if we can prove that for any sequence

{ϕ n } n∈ N in L 2 (0, T ; H), weakly convergent to some ϕ ∈ L 2 (0, T ; H), it holds lim n→∞ |u ϕn -u ϕ | E = 0.
As for condition 2, we will use Skorohod theorem and rephrase such a condition in the following way. Let ( Ω, F, P) be a probability space and let { wδ( ) (t)} t≥0 be a Wiener process, with covariance Q δ , defined on such a probability space and corresponding to the filtration { Ft } t≥0 . Moreover, let { φ } >0 and φ be { Ft } t≥0 -predictable processes taking values in S γ T , P almost surely, such that the distribution of ( φ , φ, w δ( ) ) coincides with the distribution of (ϕ , ϕ, w δ( ) ) and lim We would like to stress that condition 1 in Theorem 3.3 follows from condition 2. Actually, if we take in equation (3.6) √ = 0 and {ϕ } >0 = {ϕ n } n∈ N and ϕ deterministic, then condition 1 is a particular case of condition 2.

Proof of Theorem 3.1

In what follows, {ϕ } ∈ (0,1) and ϕ are predictable processes in A γ T , for some γ > 0 fixed, such that ϕ converges to ϕ, P almost surely, in the weak topology of L 2 (0, T ; H).

For any α ≥ 0 and > 0, we introduce the random equation

dv dt (t) = Av(t) + b(v(t) + z α (t) + Φ (t)) + α z α (t), v(0) = u 0 -z α (0), (4.1)
where z α is the process introduced in (2.6), solution of the linear equation (2.5), and

Φ (t) = t 0 e (t-s)A Q ϕ (s) ds, t ≥ 0,
is the solution of the problem

dΦ dt (t) = AΦ (t) + Q ϕ (t), Φ (0) = 0. 10
Notice that if ϕ ∈ A γ T , for some γ > 0, then

|Φ (t)| L p (D) ≤ c t 0 (t -s) -p-2 2p |ϕ (s)| H ds, so that |Φ | p L p (0,T ;L p (D)) ≤ c T 0 t 0 (t -s) -p-2 2p |ϕ (s)| H ds p dt ≤ c T |ϕ | p L 2 (0,T ;H) T 0 s -p-2 2p ds p+2 2
.

This implies that |Φ | L p (0,T ;L p (D)) ≤ c T,p √ γ, P -a.s. (4.2)
As shown e.g. in [START_REF] Flandoli | Dissipativity and invariant measures for stochastic Navier-Stokes equations[END_REF], equation (4.1) admits a unique solution

v α ∈ C([0, T ]; H) ∩ L 2 (0, T ; V ), (4.3)
and the unique generalized solution u α of equation

du(t) = [Au(t) + b(u(t)) + Q ϕ (t)] dt + √ dw δ( ) (t), t ≥ 0, u(0) = u 0 , (4.4)
can be decomposed as

u α (t) = v α (t) + z α (t) + Φ (t), t ∈ [0, T ].
Lemma 4.1. Assume that {ϕ } >0 ⊂ A γ T , for some fixed γ > 0. Then, there exists c T,γ > 0 such that for every > 0 and t ∈ [0, T ]

|v α (t)| 2 H + t 0 |v α (s)| 2 V ds ≤ c T,γ exp c |z α | 4 L 4 (0,t;L 4 (D)) |u 0 | 2 H + |z α (0)| 2 H + (α 2 + 1) |z α | 4 L 4 (0,t;L 4 (D)) + 1 . (4.5)
Moreover, we have

|v α | 4 L 4 (0,T ;L 4 (D)) ≤ c T,γ exp c |z α | 4 L 4 (0,t;L 4 (D)) |u 0 | 2 H + |z α (0)| 2 H + (α 2 + 1) |z α | 4 L 4 (0,t;L 4 (D)) + 1 2 . (4.6)
Proof. Let v α be the solution of problem (4.1), having the regularity specified in (4.3). Due to the first identity in (2.3), we have 1 2

d dt |v α (t)| 2 H + |v α (t)| 2 V = b(z α (t) + Φ (t)), v α (t) H + b(v α (t), z α (t) + Φ (t)), v α (t) H + α z α (t), v α (t) H .
For every η > 0, we have

| b(z α (t) + Φ (t)), v α (t) H | = | b(z α (t) + Φ (t), v α (t)), z α (t) + Φ (t) H | ≤ |v α (t)| V |z α (t) + Φ (t)| 2 L 4 (D) ≤ η |v α (t)| 2 V + c η |z α (t)| 4 L 4 (D) + |Φ (t)| 4 L 4 (D) .
As H 1/2 (D) → L 4 (D), by interpolation, we have

| b(v α (t), z α (t) + Φ (t)), v α (t) H | = | b(v α (t)), z α (t) + Φ (t) H | ≤ c|v α (t)| V |v α (t)| H 1/2 |z α (t) + Φ (t)| L 4 (D) ≤ c|v α (t)| 3/2 V |v α (t)| 1/2 H |z α (t) + Φ (t)| L 4 (D) ≤ η |v α (t)| 2 V + c η |v α (t)| 2 H |z α (t)| 4 L 4 (D) + |Φ (t)| 4 L 4 (D)
.

Moreover, we have

α| z α (t), v α (t) H | ≤ η |v α (t)| 2 V + c η α 2 |z α (t)| 2 H -1 .
Therefore, if we pick η = 1/6, we get

d dt |v α (t)| 2 H + |v α (t)| 2 V ≤ c |v α (t)| 2 H |z α (t)| 4 L 4 (D) + |Φ (t)| 4 L 4 (D) + c (α 2 + 1) |z α (t)| 4 L 4 (D) + c |Φ (t)| 4 L 4 (D) .
Due to (4.2), by using the Gronwall lemma this yields (4.5).

In order to prove (4.6), we notice that, as H 1/2 (D) → L 4 (D), by interpolation we have

|v α | 4 L 4 (0,T ;L 4 (D)) ≤ c T 0 |v α (s)| 2 V |v α (s)| 2 H ds ≤ |v α | 2 L ∞ (0,T ;H) |v α | 2 L 2 (0,T ;V ) .
Therefore, (4.6) follows immediately from (4.5).

Remark 4.2.

1. Due to (A.8), there exist κ ≥ 1 and c(T ) > 0 such that for any > 0

α := c(T ) |K (4, β η )| κ ∨ 1 =⇒ |z α | L 4 (0,T ;L 4 (D)) ≤ 1 and |z α (0)| H ≤ 1. (4.7)
Thanks to (4.6), this implies that

|v α | L 4 (0,T ;L 4 (D)) ≤ c T,γ |u 0 | H + α 2 + 1 , P -a.s. (4.8)
and in view of (A.9), we can conclude that if (3.3) holds, then

E |v α | κ L 4 (0,T ;L 4 (D)) ≤ c γ (T, κ) (|u 0 | κ H + 1) , κ ≥ 1. (4.9)
2. As a consequence of (4.5), if ϕ ∈ A γ T and v ϕ is a solution to the problem

dv dt (t) = Av(t) + b(v(t) + Γ(ϕ)(t)), v(0) = u 0 ,
where Γ(ϕ)(t) := t 0 e (t-s)A ϕ(s) ds, we have

|v ϕ (t)| 2 H + t 0 |v ϕ (s)| 2 H ds ≤ c T,γ 1 + |u 0 | 2 H . (4.10)
Moreover, by interpolation,

|v ϕ | L 4 (0,T ;L 4 (D)) ≤ c T,γ (1 + |u 0 | H ). (4.11)
In the next lemma we investigate the continuity properties of the operator Γ and we prove the convergence of Φ to Γ(ϕ) in case the sequence {ϕ } >0 is weakly convergent to ϕ. Lemma 4.3. For every ρ < 1, there exist θ ρ > 0 and c θ > 0 such that

|Γ(ϕ)| C θρ ([0,T ];H ρ (D)) ≤ c ρ |ϕ| L 2 (0,T ;H) , P -a.s. (4.12)
for every ϕ ∈ L 2 (0, T ; H). In particular, if {ϕ } >0 is a family in A γ T , weakly convergent in L 2 (0, T ; H) to some ϕ ∈ A γ T , for every ρ < 1 we have lim Proof. For every β ∈ (0, 1), we have

Γ(ϕ)(t) = c β t 0 (t -s) -β+1 e (t-s)A Y β (ϕ)(s) ds, where Y β (ϕ)(s) = s 0 (s -σ) -β e (s-σ)A ϕ(σ) dσ.
Due to the Young inequality, we get

|Y β (ϕ)| p L p (0,T ;H) = T 0 s 0 (s -σ) -β |ϕ(σ)| H dσ p ds ≤ |ϕ| p L 2 (0,T ;H) T 0 s -2βp p+2 ds p+2 2
, and hence, if β < 1/2 + 1/p, we have

|Y β (ϕ)| L p (0,T ;H) ≤ c p (T ) |ϕ| L 2 (0,T ;H) .
Now, as shown e.g. in [START_REF] Da Prato | Ergodicity for infinite-dimensional systems[END_REF], if β > ρ/2 + 1/p we have that the mapping

Y ∈ L p (0, T ; H) → t 0 (t -s) -β+1 e (t-s)A Y (s) ds ∈ C β-ρ 2 -1 p ([0, T ]; H ρ (D)),
is continuous. Therefore, we can conclude that

|Γ(ϕ)| C β- ρ 2 -1 p ([0,T ];H ρ (D))
≤ c ρ,β (T ) |ϕ| L 2 (0,T ;H) , P -a.s.

if ρ/2 + 1/p < β < 1/2 + 1/p, and this implies (4.12). Now, in order to prove (4.13), we notice that

Φ -Γ(ϕ) = Γ(Q (ϕ -ϕ)) + Γ(Q ϕ -ϕ).
Since Q (ϕ -ϕ) ∈ A γ T and Q (ϕ -ϕ) 0, as ↓ 0, weakly in L 2 (0, T ; H), due to the compactness of the immersion of

C θρ 1 ([0, T ]; H ρ 1 (D)) into C([0, T ]; H ρ 2 (D)), for every ρ 1 > ρ 2 , from (4.12) we conclude that lim →0 |Γ(Q (ϕ -ϕ))| C([0,T ];H ρ (D)) = 0, P -a.s. (4.14)
for every ρ < 1. Moreover, thanks again to (4.12),

|Γ(Q ϕ -ϕ)| p C([0,T ];H ρ (D)) ≤ c ρ |Q ϕ -ϕ| L 2 (0,T ;H) → 0, P -a.s.
as → 0, and together with (4.14), this implies (4.13).

Remark 4.4. Notice that, as the sequence {ϕ } >0 and the process ϕ are in A γ T , we can conclude that the convergence in (4.13) is in L p (Ω), for any p ≥ 1.

In what follows, we shall denote

ρ α (t) := v α (t) -v ϕ (t), t ≥ 0.
It is immediate to check that ρ α is a solution to the problem

dρ α dt (t) = Aρ α (t) + b(v α (t) + z α (t) + Φ (t)) -b(v ϕ (t) + Γ(ϕ)(t)) + α z α (t), ρ α (0) = -z α (0).
|ρ α (t)| 2 H + T 0 |ρ α (t)| 2 V dt ≤ c γ (T ) exp u 0 | 4 H + 1 |z α (0)| 2 H + |z α | 2 L 4 (0,T ;L 4 (D)) |v α | 2 L 4 (0,T ;L 4 (D)) + 1 + α 2 + |z α | 4 L 4 (0,T ;L 4 (D)) +|Φ -Γ(ϕ)| 2 L 4 (0,T ;L 4 (D)) 1 + |u 0 | 2 H + |v α | 2 L 4 (0,T ;L 4 (D))
.

(4.16)

Proof. Taking into account of the first identity in (2.3), we have 1 2

d dt |ρ α (t)| 2 H + |ρ α (t)| 2 V = b(v α (t)) -b(v ϕ (t)), ρ α (t) H + b(Φ (t)) -b(Γ(ϕ)(t)), ρ α (t) H + b(z α (t)), ρ α (t) H + b(v α (t), z α (t)) + b(z α (t), v α (t)), ρ α (t) H + b(z α (t), Φ (t)) + b(Φ (t), z α (t)), ρ α (t) H + b(ρ α (t), Φ (t)), ρ α (t) H + b(v ϕ (t), Φ (t) -Γ(ϕ)(t)) + b(Φ (t) -Γ(ϕ)(t), v α (t)), ρ α (t) H + α z α (t), ρ α (t) H := 8 j=1 I α ,j (t).
Now, we are going to estimate each one of the terms I α ,j (t), for j = 1, . . . , 8. We have

I α ,1 (t) = b(ρ α (t), v ϕ (t), ρ α (t) H = -b(ρ α (t)), v ϕ (t) H ,
so that, by interpolation, for any η > 0,

|I α ,1 (t)| ≤ |ρ α (t)| V |ρ α (t)| L 4 (D) |v ϕ (t)| L 4 (D) ≤ η |ρ α (t)| 2 V + c η |ρ α (t)| 2 H |v ϕ (t)| 4 L 4 (D) . (4.17)
For I α ,2 (t) we have

I α ,2 (t) = b(Φ (t), Φ (t) -Γ(ϕ)(t)) + b(Φ (t) -Γ(ϕ)(t), Γ(ϕ)(t)), ρ α (t) H ,
and, by proceeding as for I α ,1 (t), we have

|I α ,2 (t)| ≤ η |ρ α (t)| 2 V + c η |Φ (t)| 2 L 4 (D) + |Γ(ϕ)(t)| 2 L 4 (D) |Φ (t) -Γ(ϕ)(t)| 2 L 4 (D) . ( 4

.18)

For I α ,3 (t), we have

|I α ,3 (t)| = | b(z α (t)), ρ α (t) H | ≤ η |ρ α (t)| 2 V + c η |z α (t)| 4 L 4 (D) , (4.19)
and, in an analogous way,

|I α ,4 (t)| + |I α ,5 (t)| ≤ η |ρ α (t)| 2 V + c η |z α (t)| 2 L 4 (D) |v α (t)| 2 L 4 (D) + |Φ (t)| 2 L 4 (D) . ( 4

.20)

Concerning I α ,6 (t), by interpolation we get

|I α ,6 (t)| ≤ η |ρ α (t)| 2 V + c η |Φ (t)| 4 L 4 (D) |ρ α (t)| 2 H . (4.21)
Finally, with the same arguments used for I α ,3 , and also for I α ,4 and I α ,5 , we get

|I α ,7 (t)| ≤ η |ρ α (t)| 2 V + c η |v ϕ | 2 L 4 (D) + |v α (t)| 2 L 4 (D) |Φ (t) -Γ(ϕ)| 2 L 4 (D) . (4.22)
For the last term, we have

|I α ,8 (t)| ≤ η |ρ α (t)| 2 V + c η α 2 |z α (t)| 2 H -1 . (4.23)
Therefore, if we take η = 1/14, we obtain

d dt |ρ α (t)| 2 H + |ρ α (t)| 2 V ≤ c |ρ α (t)| 2 H |v ϕ (t)| 4 L 4 (D) + |Φ (t)| 4 L 4 (D) + |Φ (t)| 2 L 4 (D) + |Γ(ϕ)(t)| 2 L 4 (D) + |v ϕ (t)| 2 L 4 (D) + |v α (t)| 2 L 4 (D) |Φ (t) -Γ(ϕ)(t)| 2 L 4 (D) +c |z α (t)| 2 L 4 (D) |v α (t)| 2 L 4 (D) + |Φ (t)| 2 L 4 (D) + α 2 + c |z α (t)| 4 L 4 (D) .
Recalling that

ϕ ∈ A γ T =⇒ |Γ(ϕ)| L p (0,T ;L p (D)) ≤ c p (T )
|ρ α (t)| 2 H + T 0 |ρ α (t)| 2 V dt ≤ c γ (T ) exp |v ϕ | 4 L 4 (0,T ;L 4 (D)) |z α (0)| 2 H + |z α | 2 L 4 (0,T ;L 4 (D)) |v α | 2 L 4 (0,T ;L 4 (D)) + 1 + α 2 + |z α | 4 L 4 (0,T ;L 4 (D)) +|Φ -Γ(ϕ)| 2 L 4 (0,T ;L 4 (D)) 1 + |v ϕ | 2 L 4 (0,T ;L 4 (D)) + |v α | 2 L 4 (0,T ;L 4 (D))
.

Thanks to (4.11), we conclude that (4.16) holds.

Conclusion of the proof of Theorem 3.1

We have already seen that, if α is any given non-negative constant and v α (t) is the solution to problem (4.1), then it holds

u ϕ (t) = v α (t) + z α (t) + Φ (t), t ≥ 0. Since u ϕ (t) = v ϕ (t) + Γ(ϕ)(t)
, this implies that we can write

u ϕ (t) -u ϕ (t) = [v α (t) -v ϕ (t)] + z α (t) + [Φ (t) -Γ(ϕ)(t)] , t ≥ 0,
where α is the random constant defined in (4.7). Due to (4.8) and (4.16), it is immediate to check that

|v α (t) -v ϕ (t)| 2 H ≤ c γ (T, |u 0 | H ) |z α (0)| 2 H + |z α | 4 L 4 (0,T ;L 4 (D)) + |z α | 2 L 4 (0,T ;L 4 (D)) + |Φ -Γ(ϕ)| L 4 (0,T ;L 4 (D)) 1 + α 2 .
Now, in view of (A.8), for any β ∈ (0, 1/4) there exists c β (T ) such that for every α > 0

|z α | C([0,T ];L 4 (D)) ≤ c β (T ) K (4, β), P -a.s.
This implies that, if we fix any η ∈ (0, 1/2γ) satisfying (3.3) and β η ∈ (0, 1/4) so that (A.9) holds, we get

|v α (t) -v ϕ (t)| 2 H ≤ c γ,η (T, |u 0 | H ) K 4 (4, β η ) + K 2 (2, β) + |Φ -Γ(ϕ)| L 4 (0,T ;L 4 (D)) 1 + α 2 .
(4.24) As a consequence of (A.9) and assumption (3.3), we have sup

∈ (0,1) E α κ < ∞, κ ≥ 1.
Then, thanks again to (A.9), from (4.24) we can conclude that for any κ ≥ 1

E |v α -v ϕ | κ C([0,T ];H) ≤ c γ,η,κ (T, |u 0 | H ) δ( ) -η cκ + E |Φ -Γ(ϕ)| κ L 4 (0,T ;L 4 (D)) 1 2
.

Because of (3.3), (4.12) and (4.13), this implies that

lim →0 δ( ) -η = 0 =⇒ lim →0 E |v α -v ϕ | κ C([0,T ];H) = 0, κ ≥ 1. (4.25) Since |u ϕ -u ϕ | C([0,T ];H) ≤ |v α -v ϕ | C([0,T ];H) + |z α | C([0,T ];H) + |Φ -Γ(ϕ)| C([0,T ];H) , (4.25)
, together once more with (4.12) and (4.13), implies that

lim →0 δ( ) -η = 0 =⇒ lim →0 E |u ϕ -u ϕ | κ C([0,T ];H) = 0, κ ≥ 1. (4.26)
In view of Theorem 3.3 and all comments in Section 3 after Theorem 3.3, we can conclude that Theorem 3.1 is proved.

Proof of Theorem 3.2

In what follows, we fix any σ < 0 and p ≥ 2 such that

σ > - 2 p ∨ 2 p -1 .
Because of such a condition, we can fix two real constants α and β such that

2 p > α > -σ > 0, p ≥ 2, β ≥ 2, - 1 2 + 1 p < α 2 - 1 β < σ 2 . ( 5.1)
Once fixed α, σ, p and β, for any 0 ≤ s < t we denote

E s,t := C([s, t]; B σ p (D)) ∩ L β (s, t; B α p (D)).
E s,t turns out to be a Banach space, endowed with the norm

|v| Es,t := sup r∈ [s,t] |v(r)| B σ p (D) + |v| L p (s,t;B α p (D) .
In the case s = 0, we shall set E 0,t = E t .

Our purpose here is to show that under condition (3.2) the family {u } ∈ (0,1) satisfies a large deviation principle in C([0, T ]; B σ p (D)), with action functional I T , as defined in (3.4). In view of Theorem 3.3 and the arguments in Section 3, this follows once we prove that for any sequence {ϕ } >0 ⊂ A γ T , P-almost surely convergent to some ϕ ∈ A γ T , with respect to the topology of weak covergence in L 2 (0, T ; H), the sequence {u ϕ } >0 converges P-almost surely to u ϕ in C([0, T ]; B σ p (D)). For any > 0, we introduce the random equation

dv dt (t) = Av (t) + b(v (t) + z (t)) + Q ϕ , v (0) = u 0 -z (0), (5.2)
where z (t) = z 0 (t) is the process introduced in (2.7). In particular, we have

u ϕ (t) -u ϕ (t) = [v (t) -u ϕ (t)] + z (t) =: ρ (t) + z (t), t ≥ 0, Since dρ dt (t) = Aρ (t) + b(v (t) + z (t)) -b(u ϕ (t)) + Q ϕ (t) -ϕ(t), ρ (0) = -z (0),
we have that ρ (t) solves the following integral equation

ρ (t) = -e tA z (0) + t 0 e (t-s)A (b(v (s)) -b(u ϕ (s))) ds + t 0 e (t-s)A b(z (s)) ds + t 0 e (t-s)A (b(ρ (s), z (s)) + b(z (s), ρ (s))) ds + t 0 e (t-s)A (b(u ϕ (s), z (s)) + b(z (s), u ϕ (s))) ds + [Φ (t) -Γ(ϕ)(t)] =: 6 i=1 I ,i (t).
Our first goal here is to estimate the norm of each term I ,i in the space E t , for every t ≤ T , and prove a uniform bound for ρ in E T . To this purpose, we first prove a suitable bound for u ϕ in H θ (D), with θ ∈ (0, 1). Lemma 5.1. Assume that u 0 ∈ H θ (D), for some θ ∈ [0, 1). Then, for any ϕ ∈ L 2 (0, T ; H) we have

sup t∈ [0,T ] |u ϕ (t)| 2 H θ (D) + T 0 |u ϕ (s)| 2 H θ+1 (D) ds ≤ c |u 0 | H θ (D) , |ϕ| L 2 (0,T ;H) . (5.3)
Proof. Since 1 2 d dt |u ϕ (t)| 2 H + |u ϕ (t)| 2 V = ϕ(t), u ϕ (t) H ,
we immediately have

|u ϕ (t)| 2 H + t 0 |u ϕ (s)| 2 V ds ≤ |u 0 | 2 H + λ 1 2 t 0 |ϕ(s)| 2 H ds. (5.4) 1 2 d dt |u ϕ (t)| 2 H θ (D) + |u ϕ (t)| 2 H θ+1 (D) = b(u ϕ (t)), (-A) θ u ϕ (t) H + ϕ(t), (-A) θ u ϕ (t) H .
Now, if we assume θ < 1 and set q 1 = 2/(1 -θ) and q 2 = 2/θ, we have

b(u ϕ (t)), (-A) θ u ϕ (t) H ≤ |u ϕ (t)| L q 1 (D) |(-A) θ u ϕ (t)| L q 2 (D) |u ϕ (t)| V . As W θ,2 (D) → L q 1 (D), W 1-θ,2 (D) → L q 2 (D), this implies that b(u ϕ (t)), (-A) θ u ϕ (t) H ≤ |u ϕ (t)| H θ (D) |u ϕ (t)| H 1+θ (D) |u ϕ (t)| V ≤ 1 4 |u ϕ (t)| 2 H 1+θ (D) + c |u ϕ (t)| 2 H θ (D) |u ϕ (t)| 2 V .
Therefore, as

ϕ(t), (-A) θ u ϕ (t) H ≤ |ϕ(t)| H |u ϕ (t)| H 2θ (D) ≤ 1 4 |u ϕ (t)| 2 H 1+θ (D) + c |ϕ| 2 H ,
we conclude that

d dt |u ϕ (t)| 2 H θ (D) + |u ϕ (t)| 2 H θ+1 (D) ≤ c |u ϕ (t)| 2 H θ (D) |u ϕ (t)| 2 V + c |ϕ| 2 H .
Thanks to (5.4), this implies

|u ϕ (t)| 2 H θ (D) ≤ exp c T 0 |u ϕ (s)| 2 V ds |u 0 | 2 H θ (D) + c |ϕ| 2 L 2 (0,T ;H) ≤ exp c|u 0 | 2 H + c |ϕ| 2 L 2 (0,T ;H) |u 0 | 2 H θ (D) + c |ϕ| 2 L 2 (
|e sA z (0)| B σ p (D) + t 0 |e sA z (0)| β B α p (D) ds 1 β
, according to (5.1), for any t ≤ T we have

|I ,1 (t)| Et ≤ c |z (0)| B σ p (D) + c t 0 s -1 2 (α-σ)β ds 1 β |z (0)| B σ p (D) ≤ c T |z (0)| B σ p (D) .
(5.5)

Now, for any two processes u(t) and v(t), we define Λ(u, v)(t) := t 0 e (t-s)A b(u(s), v(s)) ds, t ≥ 0.

By proceeding as in [8, proof of Lemma 6.3], it is possible to show that if v 1 and v 2 are measurable mappings defined on [0, T ], with values in B α p (D) and B σ p (D), respectively, then

|Λ(v i , v j )(t)| B σ p (D) ≤ c t 0 (t -s) -1 2 (1+ 2 p -α) |v 1 (s)| B α p (D) |v 2 (s)| B σ p (D) ds, t ≤ T, (5.6)
and

|Λ(v i , v j )(t)| B α p (D) ≤ c t 0 (t -s) -1 2 (1+ 2 p -σ) |v 1 (s)| B α p (D) |v 2 (s)| B σ p (D) ds, t ≤ T, (5.7)
both for (i, j) = (1, 2) and for (i, j) = (2, 1).

It is immediate to check that b(v (t)) -b(u ϕ (t)) = b(ρ (t)) + b(ρ (t), u ϕ (t)) + b(u ϕ (t), ρ (t)), t ≥ 0,
so that, thanks to (5.1), from (5.6) and (5.7) we get

|I ,2 | Et ≤ c 1 (t)|ρ | L β (0,t;B α p (D)) |ρ | C([0,t];B σ p (D)) + |u ϕ | C([0,t];B σ p (D)) , t ≥ 0,
for some continuous increasing function c 1 (t), such that c 1 (0) = 0. Since we are assuming that θ ≥ σ + 1 -2/p, we have that H θ (D) → B σ p (D), so that from (5.3) we obtain

|I ,2 | Et ≤ c 1 (t) c γ (|u 0 | H θ (D) )|ρ | Et |ρ | C([0,t];B σ p (D)) + 1 . (5.8)
Concerning I ,3 (t), we first notice that b(z (t)) = div (z (t) ⊗ z (t)) = div z (t) ⊗ z (t) -ϑ δ( ) I , t ≥ 0, where ϑ δ( ) is the constant defined in (A.11), for δ = δ( ). Then, since for every ρ ≥ -1, η ≥ 0 and p ≥ 2 we have

|e tA x| B ρ p (D) ≤ c t -(1+ ρ 2 -1 p + η 2 ) |x| H -(1+η) (D) , t > 0, from (2.2) we get |I ,3 (t)| B σ p (D) ≤ c t 0 (t -s) -(1+ σ 2 -1 p + η 2 ) div(z (s) ⊗ z (s) -ϑ δ( ) I) [H -(1+η) (D)] 4 ds ≤ c t 0 (t -s) -(1+ σ 2 -1 p + η 2 ) z (s) ⊗ z (s) -ϑ δ( ) I) [H -η (D)] 4 ds.
In the same way, we have

|I ,3 (t)| B α p (D) ≤ c t 0 (t -s) -(1+ α 2 -1 p + η 2 ) z (s) ⊗ z (s) -ϑ δ( ) I) [H -η (D)] 4 ds.
Due to (5.1), this implies that we can find η > 0 and ρ ≥ 1 such that

|I ,3 | Et ≤ c 2 (t) z ⊗ z -ϑ δ( ) I L ρ (0,T ;[H -(1+γ) (D)] 4 ) .
(5.9)

For I ,4 (t), by using again (5.6) and (5.7), we have

|I ,4 (t)| B σ p (D) ≤ c t 0 (t -s) -1 2 (1+ 2 p -α) |ρ (s)| B α p (D) |z (s)| B σ p (D) ds, and
|I ,4 (t)| B α p (D) ≤ c t 0 (t -s) -1 2 (1+ 2 p -σ) |ρ (s)| B α p (D) |z (s)| B σ p (D) ds,
and then, according to (5.1), we can find ρ ≥ 1 such that

|I ,4 | Et ≤ c 3 (t) |ρ | L β (0,t;B α p (D)) |z | L ρ (0,T ;B σ p (D)) ≤ c 3 (t) |ρ | Et |z | L ρ (0,T ;B σ p (D)) .
(5.10)

As for I ,4 (t), for I ,5 (t) we have

|I ,5 (t)| B σ p (D) ≤ c t 0 (t -s) -1 2 (1+ 2 p -α) |u ϕ (s)| B α p (D) |z (s)| B σ p (D) ds, and
|I ,5 (t)| B α p (D) ≤ c t 0 (t -s) -1 2 (1+ 2 p -σ) |u ϕ (s)| B α p (D) |z (s)| B σ p (D) ds.
As are assuming θ ≥ σ + 1 -2/p, we have that θ > α -2/p, so that for any η > 0 such that θ -η > α -2/p, we have H 1+θ-η (D) → B α p (D). By interpolation, this implies

|x| B α p (D) ≤ c η |x| H 1+θ-η (D) ≤ c η |x| 1-η H 1+θ (D) |x| η H θ (D) , so that |x| 2 1-η B α p (D) ≤ c η |x| 2 H 1+θ (D) |x| 2η 1-η H θ (D) .
According to (5.3), this implies that u ϕ ∈ L Due to condition (5.1), since θ ≥ σ + 1 -2/p, we can find η ∈ (0, 1) such that

1 - 2 β < η < θ + 2 p -α.
For such η > 0 we have

|I ,5 (t)| B σ p (D) ≤ c t 0 s -1 2 (1+ 2 p -α) β β-1 ds β-1 β |u ϕ | L 2 1-η (0,t;B α p (D)) |z | L κ (0,t;B σ p (D)) , (5.12)
where

1 κ = 1 - 1 -η 2 + β -1 β = 1 β - 1 -η 2 .
Analogously, if we pick η > 1 -2/p, we get

t 0 |I ,5 (s)| β B α p (D) ds ≤ c t 0 s -1 2 (1+ 2 p -σ) ds β |u ϕ | β L 2 1-η (0,t;B α p (D)) |z | L 2β 2-β(1-η) (0,t;B σ p (D))
.

This implies that if we take t 0 > 0 such that

3 c(t 0 ) c γ (|u 0 | H θ (D) ) ≤ 1 2 , for any n ≥ n 0 and t ≤ τ n (ω) ∧ t 0 |ρ n (ω)| Et ≤ c |Φ n (ω) -Γ(ϕ)(ω)| E T +c T |z n (ω)| C([0,T ];B σ p (D)) + z n (ω) ⊗ z n (ω) -n ϑ δ( ) I L ρ (0,T ;[H -γ (D)] 4 ) .
As a consequence of (5.17), there exists

n 1 = n 1 (ω) ≥ n 0 such that c T |z n (ω)| C([0,T ];B σ p + z n (ω) ⊗ z n (ω) -n ϑ δ( ) I L ρ (0,T ;[H -γ (D)] 4 ) +c |Φ n (ω) -Γ(ϕ)(ω)| E T ≤ 1 2 , n ≥ n 1 ,
so that τ n (ω) ∧ t 0 = t 0 , for n ≥ n 1 , and

|ρ n (ω)| Et 0 ≤ c |Φ n (ω) -Γ(ϕ)(ω)| E T +c T |z n (ω)| C([0,T ];B σ p (D)) + z n (ω) ⊗ z n (ω) -n ϑ δ( ) I L ρ (0,T ;[H -γ (D)] 4 ) .
Now, we can repeat the same argument in the intervals [(i -1)t 0 , it 0 ], for i = 0, . . . , i T , where i T is the smallest integer such that i T t 0 ≥ T , and we find

|ρ n (ω)| E (i-1)t 0 ,it 0 ≤ i c |Φ n (ω) -Γ(ϕ)(ω)| E T +i c T |z n (ω)| C([0,T ];B σ p (D)) + z n (ω) ⊗ z n (ω) -n ϑ δ( ) I L ρ (0,T ;[H -γ (D)] 4 ) , (5.18)
for every n ≥ n i = n i (ω), where n i (ω)

≥ n i-1 (ω) is such that c T |z n (ω)| C([0,T ];B σ p (D)) + z n (ω) ⊗ z n (ω) -n ϑ δ( ) I L ρ (0,T ;[H -γ (D)] 4 ) +c |Φ n (ω) -Γ(ϕ)(ω)| E T ≤ 1 2i , n ≥ n i .
Therefore, from (5.18) we obtain that for any n ≥ n i T (ω)

|ρ n (ω)| E T ≤ i c |Φ n (ω) -Γ(ϕ)(ω)| E T +i c T |z n (ω)| C([0,T ];B σ p (D)) + z n (ω) ⊗ z n (ω) -n ϑ δ( ) I L ρ (0,T ;[H -γ (D)] 4 ) ,
and due to (5.17

A Appendix

Here we describe and prove some properties of the solution of the linear problem. As in Section 2, for every α ≥ 0 and > 0 we denote by z α (t) the stationary solution of the linear problem

dz(t) = (A -α)z(t) dt + √ dw δ( ) (t), t ≥ 0.
The process z α (t) is given by

z α (t) = √ t -∞ e (t-s)(A-α) d wδ( ) (s), t ≥ 0.
As we already mentioned in Section 2, for any fixed > 0 the process z α belongs to the space L p (Ω; C([0, T ]; D((-A) β )), for any T > 0, p ≥ 1 and β < γ/2. We first want to estimate the norm of z α in Besov spaces of negative exponent.

Lemma A.1. For any α ≥ 0 and > 0 and for any p, κ ≥ 1 and σ < σ < 0 it holds

E sup t∈ [0,T ] |z α (t)| κ B σ p (D) ≤ c κ,p   k∈ Z 2 0 |k| 2(σ -1)   κ 2 . (A.1) Proof. Since z α (t) = (-A) -σ 2 (-A) σ 2 z α (t), we have |z α (t)| B σ p (D) ≤ |(-A) σ 2 z α (t)| L p (D) . (A.2)
By using stochastic factorization, for any β ∈ (0, 1) we have

(-A) σ 2 z α (t) = sin πβ π t -∞ (t -s) β-1 e (t-s)A Y ,β (s) ds, where Y ,β (s) = s -∞ (s -ρ) -β e (s-ρ)A (-A) σ 2 dw δ( ) (ρ).
Therefore, if we take p ≥ 1/β, we get .

|(-A) σ 2 z α (t)| p L p (D) ≤ c β,p t -∞ s - (1-β)p p-1 e -p p-1 s ds p-1 t -∞ |Y ,β (s)| p L p (D) ds. (A.3) Now, for any t ∈ R and x ∈ D E |Y ,β (t, x)| p = c p p 2 E   k∈ Z 2 0 t -∞ λ k (δ (
The general case follows from the Hölder inequality.

Next, we estimate the norm of z α in L p (D)-spaces. This Lemma could be proved using stochastic calculus in Banach spaces and the notion of γ-radonfying operators (see [START_REF] Brzeźniak | On stochastic convolution in Banach spaces and applications[END_REF]). We give here an elementary proof. In what follows, it will be important that the random variable K (p, β) has all moments finite, with an uniform bound with respect to > 0.

Lemma A.3. Let p, q ≥ 1 and > 0 be fixed. Then, for any η ∈ (0, 1/2γ) there exists β η ∈ (0, 1/4) such that E |K (p, β η )| q ≤ c p,βη,q δ( ) -η cq,p , (A.9)

Proof. It is immediate to check that, for any q ≥ m, we have E|K (p, β)| q ≤ c p,β,q +∞ -∞

(1 + σ 2 ) -1 E |Y (σ)| q L p (D) dσ. This implies that for any p, q ≥ 1 E |Y (σ)| q L p (D) ≤ c 1 (q, p) Λ β ( ) c 2 (q,p) , Because of (A.12), this implies (A.13) in the case κ = p ≥ 1. The case κ, p ≥ 1 follows from the Hölder inequality and the fact that L p (D) ⊂ L q (D), if p ≥ q.

Proof

Theorem 3 . 1 .

 31 Let → δ( ) be a function satisfying (3.2). Moreover, assume that there exists η > 0 such that lim →0 δ( ) -η = 0. (3.3)

→0φ

  = φ weakly in L 2 (0, T ; H), P -a.s.Then, if ū φ is the solution of an equation analogous to(3.6), with ϕ and w δ( ) replaced respectively by φ and w δ( ) , we have that lim →0 ū φ = ū φ in E, P -a.s.(3.8)

→0|Φ-

  Γ(ϕ)| C([0,T ];H ρ (D)) = 0, P -a.s. (4.13)

Lemma 4 . 5 .

 45 If {ϕ } >0 ⊂ A γ T and ϕ ∈ A γ T , for every α ≥ 0 we have sup t∈ [0,T ]

2 1 - 2 1

 12 η (0, T ; B α p (D)) and|u ϕ | L -η (0,T ;B α p (D)) ≤ c γ,η (|u 0 | H θ (D) ). (5.11)

  ) we can conclude that lim n→∞ |ρ n (ω)| E T = 0.

2 , 2 .

 22 ))|k| σ (t -s) -β e -(t-s)(|k| 2 +α) e k (x) dβ k (s) δ( )) 2 |k| 2σ (t -s) -2β e -2(t-s)(|k| 2 +α) |e k (x)| 2 dsso that, integrating with respect to x ∈ D, for any β < -σ/2, and hence p ≥ -2/σ,E |Y ,β (t)| p L p (D) ≤ c pTherefore, thanks to (A.2) and (A.3), for any κ ≥ p ≥ 2/σ this yieldsE sup t∈ [0,T ] |z α (t)| κ B σ p (D) ≤ E sup t∈ [0,T ]|(-A)

σ 2 z

 2 α (t)| κ L p (D) ≤ c κ,p E sup t∈ [0,T ] |(-A)

Lemma A. 2 .e( 1 + σ 2 )

 212 For every α ≥ 0 and > 0 and for every p ≥ 1 it holdsE |z α (t)| p L p (D) ≤ c p (For every p ≥ 1 we have E |z α (t)| p L p (D) t-s)(|k| 2 +α) λ k (δ( ))e k (x) d βk (s) -2(t-s)(|k| 2 +α) λ k (δ( )) 2 |e k (x)| 2 ds implies that (A.4) holds.Now, by proceeding as in the proof of [9, Proposition 2.1], it is possible to show that for any p ≥ 1 and β ∈ (0, 1/4) there exist θ = θ(p, β) ∈ (0, 1/4) and ρ = ρ(p, β) ∈ (0, 1), and a random variable K (p, β) such that for any α ≥ 0 and > 0|z α (t)| L p (D) ≤ (α ∨ 1) -θ (1 + t ρ ) K (p, β), P -a.s.,(A.5)whereK (p, β) = c p,β +∞ -∞ -1 |Y (σ)| m L p (D) dσ s) -β e (σ-s)A dw δ( ) (s). (A.7)In particular, we have |z α | C([0,T ];L p (D)) ≤ (α ∨ 1) -θ c p (T ) K (p, β), P -a.s. (A.8)

2   k∈ Z 2 0|e k | 2 L ∞ (D) ∞ 0 s 1   p 2 =: c p Λ β ( ) p 2 .

 2220122 s) -β λ k (δ( ))e -|k| 2 (σ-s) e k (x) d βk (s),we haveE |Y (σ, x)| p ≤ c p p/-2β λ k (δ( )) 2 e -|k| 2 s ds 2β) (1 + δ( )|k| 2γ ) -

  γ, P -a.s. as a consequence of the Gronwall lemma, this implies that

	sup
	t∈ [0,T ]

  . It is immediate to check that -(t-s)|k| 2 dβ k (s).The process z(t) is stationary Gaussian and L(z(t)) = µ, for every t ∈ R. This means that for any p ≥ 1E |z ⊗ z -ϑ δ( ) I| p L p (0,T ;[H σ (D)] 4 ) = E

		z (t) =	√	Q z(t), t ∈ R,
	where	t		t
	z(t) =	e (t-s)A dw(t) =
		-∞		0 k∈ Z 2	-∞
			0	T	|z (t) ⊗ z (t) -ϑ δ( ) I| p [H σ (D)] 4 dt
	= p T

e H |Q h ⊗ Q h -ϑ δ( ) I| p [H σ (D)] 4 µ(dh) = p T H |h δ( ) ⊗ h δ( ) -ϑ δ( ) I| p [H σ (D)] 4 µ(dh).

* This material is based upon work supported by the National Science Foundation under grant No. 0932078000, while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2015 semester. † Partially supported by the NSF grant DMS 1407615. ‡ Partially supported by the French government thanks to the ANR program Stosymap and the "Investissements d'Avenir" program ANR-11-LABX-0020-01.

Thanks to (5.11), this, together with (5.12), implies that there exists some ρ ≥ 1 such that

(5.13)

Collecting together (5.5), (5.8), (5.9), (5.10) and (5.13), we conclude that

for some continuous increasing function c(t) such that c(0) = 0. Now, we are going to show that for any sequence { n } n∈ N converging to zero, there exists a subsequence

( 5.14) and this clearly implies that lim

Let { n } n∈ N be a sequence converging to zero. As we are assuming that α < 2/p, there exists ρ < 1 such that H ρ (D) → B α p (D), so that, due to (4.13) we have lim →0

|Φ -Γ(ϕ)| E T = 0, P -a.s.

(5.16)

Then, as a consequence of (A.1), (A.13) and (5.16), we have that there exists a subsequence of { n } n∈ N , that for simplicity of notations we are still denoting by { n } n∈ N , and a set Ω ⊆ Ω with

(5.17)

Next, for any > 0 we denote

If we fix any ω ∈ Ω , in view of (A.1) there exists some n 0 = n 0 (ω) ∈ N such that for any n ≥ n 0 and t ≤ τ n (ω)

for some positive constants c 1 (q, p) and c 2 (q, p). Now, we have

Therefore, if we pick any η ∈ (0, 1/2γ) and define β η := η γ/2, we have

and this implies (A.9).

In what follows, we shall denote H := R Z 2 0 and µ := N (0, (-A) -1 /2). The Gaussian measure µ is defined on H, but in fact µ(H σ (D)) = 1, if σ < 0, so that the support of µ is contained in H σ (D), for every σ < 0. Now, for any h ∈ H and δ > 0, we define

where we recall that, for any k ∈ Z 2 0 and δ > 0,

Next, for i = 1, 2 we define

where

By proceeding as in [START_REF] Da Prato | Two-dimensional Navier-Stokes equations driven by a spacetime white noise[END_REF]Appendix] it is possible to prove that for i = 1, 2

for every κ ≥ 1 and σ < 0. In particular, due to definition (A.10), this implies that ∃ lim