Random Attractors for the Stochastic Navier–Stokes Equations on the 2D Unit Sphere

In this paper we prove the existence of random attractors for the Navier–Stokes equations on 2 dimensional sphere under random forcing irregular in space and time. We also deduce the existence of an invariant measure.


Introduction
Complex three dimensional flows in the atmosphere and oceans are modelled assuming that the Earth's surface is an approximate sphere. Then it is natural to model the global atmospheric circulation on Earth (and large planets) using the Navier-Stokes equations (NSE) on 2-dimensional sphere coupled to classical thermodynamics [34]. This approach is relevant for geophysical flow modeling.
Many authors have studied the deterministic NSEs on the unit sphere. Notably, Il'in and Filatov [30,32] considered the existence and uniqueness of solutions to these equations and estimated the Hausdorff dimension of their global attractors [31]. Temam and Wang [42] considered the inertial forms of NSEs on sphere while Temam and Ziane [43], see also [4], proved that the NSEs on a 2-dimensional sphere is a limit of NSEs defined on a spherical shell [43]. In other directions, Cao et al. [14] proved the Gevrey regularity of the solution and found an upper bound on the asymptotic degrees of freedom for the long-time dynamics.
Concerning the numerical simulation of the deterministic NSEs on sphere, Fengler and Freeden [25] obtained some impressive numerical results using the spectral method, while the numerical analysis of a pseudo-spectral method for these equations has been carried out in Ganesh et al. [27].
In our earlier paper [9] we analysed the Navier-Stokes equations on the 2-dimensional sphere with Gaussian random forcing. We proved the existence and uniqueness of solutions and continuous dependence on data in various topologies. We also studied qualitative properties of the stochastic NSEs on the unit sphere in the context of random dynamical systems.
Building on those preliminary studies, in the current paper, we prove the existence of random attractors for the stochastic NSEs on the 2-dimensional unit sphere. Let us recall here that, given a probability space, a random attractor is a compact random set, invariant for the associated random dynamical system and attracting every bounded random set in its basis of attraction (see Definition 4.4).
In the area of SPDEs the notions of random and pullback attractors were introduced by Brzeźniak et al. [7], and by Crauel and Flandoli [16]. These concepts have been later used to obtain crucial information on the asymptotic behaviour of random [7], stochastic [2,16,17,26] and non-autonomous PDEs [13,33,36].
We do not know if our system is dissipative in H 1 . Therefore, despite the fact that the embedding H 1 → L 2 is compact, the asymptotic compactness approach seems to be the only method available in the L 2 -setting to yield the existence of an attractor, hence of an invariant measure.
The paper is organised as follows. In Sect. 2, we recall the relevant properties of the deterministic NSEs on the unit sphere, outline key function spaces, and recall the weak formulation of these equations. In Sect. 3, we define the stochastic NSEs on the unit sphere. The stochastic NSEs is decomposed into an Ornstein-Uhlenbeck process and a deterministic NSEs with random forcing. First we construct a stationary solution of the Ornstein-Uhlenbeck process (associated with the linear part of the stochastic NSEs) and then identify a shift-invariant subset of full measure satisfying a strong law of large numbers. We then review the key existence and uniqueness results obtained in [9]. In Sect. 4 we prove the existence of a random attractor of the stochastic NSEs on the 2-d sphere, which is the main result of the paper. In doing so, we present Lemma 4.8, which is a corrected version of [9,Lemma 6.5]. Based on the asymptotic results in the lemma, a new class of functions R is defined in Definition 4.11. The class DR of all closed and bounded random sets is then defined using functions in the class R. The main results are given in Theorem 4.14, which asserts that the random dynamical system ϕ generated by the NSEs on the unit sphere is DR-asymptotically compact. Hence, in view of a result on existence of a random attractor (Theorem 4.6), the existence of a random attractor of ϕ is deduced.
The paper is concluded with a simple proof of the existence of an invariant measure and some comments on the question of its uniqueness.
In our paper a special attention is given to the noise with low space regularity. While many works on random attractors consider only finite dimensional noise, we follow here the approach from Brzeźniak et al. [8] and consider an infinite dimensional driving Wiener process with minimal assumptions on its Cameron-Martin space (known also as the Reproducing Kernel Hilbert Space), see Remark 3.5 and the Introduction to [8] for motivation.

The Navier-Stokes Equations on a Rotating Unit Sphere
The sphere is a very special example of a compact Riemannian manifold without boundary hence one could recall all the classical tools from differential geometry developed for such manifolds. However we have decided to follow a different path of using the polar coordinates and defining all such objects directly. Our presentation here is a self-contained version of an analogous introductory section from our companion paper [9]. A reader who is familiar with the last reference can skip reading this section.

Preliminaries
By S 2 we will denote the standard 2-dimensional unit sphere, i,e, a subset of the Euclidean space R 3 described by Let us now define the surface gradient ∇ and curl operators acting on tangent vector fields and the surface gradient acting on scalar functions, all with respect to scalar product in the tangent spaces T x S 2 inherited from R 3 .
Suppose that u and v are two tangent vector fields on S 2 and f is real valued function on S 2 , all of C 1 class. By some classical results, see for instance [23] or [22,Definition 3.31], there exist a neighbourhood U of S 2 in R 3 and vector fieldsũ : U → R 3 ,ṽ : U → R 3 and a functionf :

2)
Vol. 20 (2018) Random Attractors for the Stochastic Navier-Stokes Equations 229 where∇ is the gradient in R 3 and, for x ∈ S 2 , the map π x : R 3 → T x S 2 is the orthogonal projection, i.e.
Let us point out that the definitions of (∇ v u) and curl u above are independent of the choice of the extensionsũ andṽ. In the former case, this can be shown either using a general approach from the references above or, as in our companion paper [9] by exploiting a well known formula for the R 3 -vector product 1 to get If follows from the definition (2.3) that curl u is a normal vector field on S 2 , i.e. curl u(x) ⊥ T x S 2 for every x ∈ S 2 . Since the co-dimension of T x S 2 in R 3 is equal to 1, this normal vector field can be identified with a scalar function on S 2 denoted by curl u by Note that it follows that curl u( Lemma 2.1. Ifũ andṽ are R 3 -valued vector fields on S 2 , and u and v are tangent vector fields on S 2 , defined by u(x) = π x (ũ((x))) and v(x) = π x (ṽ(x)), x ∈ S 2 , then the following identity holds Proof. Let us fix x ∈ S 2 . Then we can decompose vectorsũ =ũ(x) andũ =ṽ(x) into the tangential u = u(x) ∈ T x S 2 and v = v(x) ∈ T x S 2 , and the normal component as follows Hence the lemma is proved.
Suppose now that u is a tangent vector field on S 2 andũ is a R 3 -valued vector field defined in some neighbourhood U of S 2 in R 3 such thatũ| S 2 = u. Applying formula (2.8) to the vector fields, 2ũ and v =∇ ×ũ, since also (ũ · x) = 0 we get Hence by formulae (2.10), (2.7) and the above definitions we obtain Here, we use the following notation. Given another tangential vector field v on S 2 , we will denote by v × curl u a tangential vector field defined as the R 3 vector product of vectors, tangent v and normal curl u, i.e.
[ Thus from the above and (2.13) we infer that We will use the classical spherical coordinates to describe (in a non-unique way) the points on the sphere With a bit of ambiguity, if x = x(θ, φ) as in (2.14), the angles θ and φ will be denoted by θ(x) and φ(x), or just θ and φ.
For (θ, φ) ∈ [0, π] × [0, 2π), by e θ = e θ (θ, φ) and e φ = e φ (θ, φ) we will denote an orthonormal basis in the tangent plane T x S 2 , where x = x(θ, φ), defined by e θ = (cos θ cos φ, cos θ sin φ, − sin θ), e φ = (− sin φ, cos φ, 0). (2.15) If f : S 2 → R is C 1 a class function, then we can show that the surface gradient of f has the following form In what follows, we will often not distinguish between functions f andf and use the notation f for both. Similarly, if u is a (tangential) vector field on S 2 which can be written in a form u = (u θ , u φ ) with respect to the (moving) basis e θ , e φ , that is we define the surface divergence with respect to the surface area on S 2 by the formula With slight abuse of notation, for x ∈ S 2 , Finally, see [30], if f a scalar function on S 2 , then we define a tangent vector field Curl f by The surface diffusion operator acting on tangential vector fields on S 2 is denoted by Δ (known as the vector Laplace-Beltrami or Laplace-de Rham operator) and is defined as (2.20) Using (2.18) and (2.19), one can derive the following relations connecting the above operators: The Navier-Stokes equations (NSEs) for the evolution of the (tangential) velocity vector field u(t, x) = (u θ (t, x), u φ (t, x)) on the 2-dimensional rotating unit sphere S 2 under the influence of an external force f (x) = (f θ , f φ ) takes the following form [24,40] Let us describe the notations used above in more details. Firstly, ν and ρ are two positive constants which can be seen as simplified physical constants called the viscosity and the density of the fluid. The word "rotational" refers to the Coriolis acceleration ω which is normal vector field defined by Vol. 20 (2018) Random Attractors for the Stochastic Navier-Stokes Equations 231 where Ω is a given constant. Note that if In what follows we will identify the normal vector field ω with the corresponding scalar function ω defined by The operator L is given by [40] L = Δ + 2Ric, (2.24) where Δ is the Laplace-de Rham operator, see (2.20), and Ric denotes the Ricci tensor of the twodimensional sphere S 2 . It is well known that (see e.g. [45, p. 75 We remark that in papers in [14,30,32,42] the authors consider NSEs with L = Δ but the analysis in our paper are still valid in that case.

Function Spaces on the Sphere
In what follows we denote by dS the Lebesgue integration with respect to the surface measure (or the volume measure when S 2 is seen as a Riemannian manifold). In the spherical coordinates we have, locally, dS = sin θdθdφ. For p ∈ [1, ∞) we will use the notation L p = L p (S 2 ) for the space L p S 2 , R of p-integrable scalar functions on S 2 endowed with the norm For p = 2 the corresponding inner product is denoted by We will denote by L p = L p (S 2 ) the space L p S 2 , T S 2 of vector fields v : where, for x ∈ S 2 , |v(x)| stands for the length of v(x) in the tangent space T x S 2 . For p = 2 the corresponding inner product is denoted by Throughout the paper, the induced norm on L 2 (S 2 ) is denoted by · and for other inner product spaces, say X with inner product (·, ·) X , the associated norm is denoted by · X . We have the following identities for appropriate scalar and vector fields [30, (2.4)-(2.6)]: (Curl curl w, z) = (curl w, curl z). (2.28) In (2.27), the L 2 (S 2 ) inner product is used on the left hand side and the L 2 (S 2 ) inner product is used on the right hand side. We now introduce Sobolev spaces H s (S 2 ) = H s,2 (S 2 ) and H s (S 2 ) = H s,2 (S 2 ) of scalar functions and vector fields on S 2 respectively. Let ψ be a scalar function and let u be a vector field on S 2 , respectively. For s ≥ 0 we define where we have used formulas (2.20),(2.26)-(2.28). We note that for k = 0, 1, 2, . . . and θ ∈ (0, 1) the space H k+θ (S 2 ) can be defined as the interpolation space between H k (S 2 ) and H k+1 (S 2 ). We can apply the same procedure for H k+θ (S 2 ).
One has the following Poincaré inequality [32, Lemma 2] for some positive constant λ 1 . The space of smooth (C ∞ ) tangential fields on S 2 can be decomposed into three components, one in the space of all divergence-free fields and the others through the Hodge decomposition theorem [3, Theorem 1.72]: Since V is densely and continuously embedded into H and H can be identified with its dual H , we have the following imbeddings: (2.35) We say that the spaces V, H and V form a Gelfand triple.

The Weak Formulation
We consider the linear Stokes problem By taking the inner product of the first equation of (2.36) with v ∈ V and then using (2.28), we obtain In view of (2.31) and (2.25), the bilinear form a satisfies and hence it is continuous on V . So by the Riesz Lemma, there exists a unique operator A : Using the Poincaré inequality (2.32), we also have a(u, u) ≥ α u 2 V , with α = λ 1 − 2ν, which means a is coercive in V whenever λ 1 > 2ν. In practice, usually one has λ 1 2ν. Hence by the Lax-Milgram theorem the operator A : V → V is an isomorphism. Furthermore, by using [39, Theorem 2.2.3], we conclude that the operator A is positive definite, self-adjoint in H and D(A 1/2 ) = V .
Vol. 20 (2018) Random Attractors for the Stochastic Navier-Stokes Equations 233 Next we define an operator A in H as follows: Let P be the Leray orthogonal projection from L 2 (S 2 ) onto H. It can be shown [28] that D(A) = H 2 (S 2 ) ∩ V and A = −P (Δ + 2Ric), and A * = A. It can also be shown that V = D(A 1/2 ) and Using the following identity and (2.27), for divergence free tangential vector fields v, w, z, the trilinear form can be written as The Coriolis operator C 1 : Clearly, C 1 is linear and bounded in L 2 S 2 . In the sequel we will need the operator C = P C 1 which is well defined and bounded in H. Furthermore, for u ∈ H This weak formulation can be written in operator equation With a slight abuse of notation, we also denote B(u) = B(u, u).
The following are some fundamental properties of the trilinear form b; see [25]: there exists a constant C > 0 such that (2.47) We also need the following estimates: In view of (2.50), b is a bounded trilinear map from L 4 (S 2 ) × V × L 4 (S 2 ) to R. Moreover, we have the following result: It can be seen from (2.50) that b is a bounded trilinear map from L 4 (S 2 ) × V × L 4 (S 2 ) to R. It follows that B maps L 4 (S 2 ) ∩ H (and so V ) into V and by using the following inequality from [32, p. 12]

Preliminaries
Let us recall that for a real separable Hilbert space K and a real separable Banach space X, a linear operator U : Here γ K is the canonical Gaussian cylindrical measure on K. If a linear map U : K → X is γ−radonifying, then γ K • U −1 has a unique extension to a Borel probability measure denoted by ν U on X. By R(K, X) we denote the Banach space of γ-radonifying operators from K to X with the norm From now on we will freely use notation introduced in the former sections. It follows from [12, Theorem 2.3] that for a self adjoint operator U ≥ cI in H, where c > 0, such that U −1 is compact, the operator where {e } is an orthonormal basis of H corresponding to U . This implies the following result.
Proof. Let us recall the following well know facts.
where the RHS is independent of x ∈ S 2 . (iii) If P being the Legendre polynomial of degree , then P (1) = 1. Therefore, condition (3.1) yields if and only if s > 1 2 and the lemma follows. Let Remark 3.2. It follows from Lemma 3.1 that if s > 1/2 then the operator Let us recall, that X is an M -type 2 Banach space, see [6] for details. The Stokes operator −A restricted to X is an infinitesimal generator of an analytic semigroup. We will consider an operator in X defined by the formulâ where ν > 0, and C is the Coriolis operator. For the reader's convenience we recall a result presented in [9].  Let us denote by C 1/2 (R, X) the space of all continuous functions ω : R → X such that The space C 1/2 (R, E) endowed with the norm · C 1/2 (R,E) is a nonseparable Banach space. We denote by F the Borel σ-algebra on Ω(ξ, E). One can show [5] that for ξ ∈ (0, 1/2), there exists a Borel probability measure P on Ω(ξ, E) such that the canonical process w t , t ∈ R, defined by is a two-sided Wiener process. The Cameron-Martin (or Reproducing Kernel Hilbert space) of the Gauss- . Moreover, the family (W t ) t∈R is an H-cylindrical Wiener process on a filtered probability space (Ω(ξ, E), F, P), where F = F t ) t∈R in the sense of e.g. [11].

Ornstein-Uhlenbeck Process
The following is our standing assumption.
On the space Ω(ξ, E) we consider a flow ϑ = (ϑ t ) t∈R defined by This was an assumption in [10, Proposition 6.2]. Rewriting that proposition in a slightly more general form we have Proposition 3.6. For any α ≥ 0, the operator −(Â + αI) is a generator of an analytic semigroup is well defined, linear and bounded. In particular, there exists a constant C < ∞ such that for (3.9) The following results for the operatorÂ follow from Corollary 6.4, Theorem 6.6 and Corollary 6.8 in from [10], respectively.
Thus, τ s is a linear a bounded map from C 1/2 (R, X) into itself. Moreover, the family (τ s ) s∈R is a C 0 group on C 1/2 (R, X). Using this notation Theorem 3.8 can be rewritten in the following way.
i.e. for any t ≥ 0, For ω ∈ C ∞ 0 (R) with ω(0) = 0, by the fundamental theorem of calculus, we obtain whereω(t) = dω(t)/dt. Hence z α (t) is the solution of the following equation It follows from Theorem 3.8 that We can view the formula (3.10) as a definition of a process z α (t), t ∈ R, on the probability space (Ω(ξ, E), F, P). Equation (3.11) suggests that this process is an Ornstein-Uhlenbeck process.
Proposition 3.10. The process z α (t), t ∈ R, is a stationary Ornstein-Uhlenbeck process. It is the solution of the equation i.e. for all t ∈ R, a.s.

13)
where the integral is the Itô integral on the M -type 2 Banach space X in the sense of [6]. In particular, for some constant C depending on X, Moreover, E z α (t) 4 X tends to 0 as α → ∞. Proof. Stationarity of the process z α follows from Eq. (3.12). The equality (3.13) follows by finitedimensional approximation.
By the Burkholder inequality, see [6] and [35], we have Hence, we conclude that the last integral (3.15) is finite. Finally, the last statement follows from (3.15) by applying the Lebesgue Dominated Convergence Theorem.
By Proposition 3.10, z α (t), t ∈ R, is a stationary and ergodic X-valued process, hence by the Strong Law for Large Numbers (see Da Prato and Zabczyk [21] for a similar argument), It also follows from Proposition 3.10 that we can find α 0 such that for all α ≥ α 0 , where λ 1 is the constant appearing in the Poincaré inequality (2.32) and C > 0 is a certain universal constant. By adding a white noise term to (2.22) the stochastic NSEs on the sphere is where we assume that u 0 ∈ H, f ∈ V and n(t, x) is a Gaussian random field which is a white noise in time. In the same way as in the deterministic case we apply the operator of projection onto the space of divergence free fields and reformulate the above equation as an Itô type equation Here f is the deterministic forcing term and u 0 is the initial velocity. We assume that W is a cylindrical Wiener process on a certain Hilbert space K defined on a probability space (Ω, F, P), see [20] and [11]. G is a linear continuous operator from K to H. The space K, which is the RKHS of the Wiener process, determines the spatial smoothness of the noise term, will satisfy further assumptions to be specified later.
We have proved the following major theorems on the existence and uniqueness of the solution of (3.21)-(3.22) in [9].

A measurable dynamical system (DS) is a triple
where (Ω, F) is a measurable space and ϑ : R × Ω (t, ω) → ϑ t ω ∈ Ω is a measurable map such that for all t, s ∈ R, ϑ t+s = ϑ t • ϑ s . A metric DS is a quadruple where (Ω, F, P) is a probability space and (Ω, F, ϑ) is a measurable DS such that for each t ∈ R, ϑ t : Ω → Ω preserves P. Denote by Ω α (ξ, E) the set of those ω ∈ Ω(ξ, E) for which the equality (3.17) holds true. It follows from Corollary 3.9 that this set is invariant with respect to the flow ϑ, i.e. for all α ≥ 0 and all t ∈ R, ϑ t (Ω α (ξ, E)) ⊂ Ω α (ξ, E). Therefore, the same is true for a set It follows that as a model for a metric dynamical system we can take either the quadruple (Ω(ξ, E), F, P, ϑ) or the quadruple (Ω(ξ, E),F ,P,θ), whereF,P, andθ are respectively the natural restrictions of F, P and ϑ toΩ(ξ, E). Suppose also that (X, d) is a Polish space (i.e. complete separable metric space) and B is its Borel σ−field. Let R + = [0, ∞). ϕ(s, ω) for all s, t ∈ R + and ϕ(0, ω) = id, for all ω ∈ Ω. (Cocycle property) An RDS ϕ is said to be continuous or differentiable iff for all (t, ω) ∈ R + × Ω, ϕ(t, ·, ω) : X → X is continuous or differentiable, respectively. Similarly, an RDS ϕ is said to be time continuous iff for all ω ∈ Ω and for all x ∈ X, ϕ(·, x, ω) : R + → X is continuous.

Definition 4.2. Given a metric DS T and a Polish space X, a map
For two nonempty sets A, B ⊂ X, we put In fact, ρ restricted to the family CB of all nonempty closed subsets on X is a metric, and it is called the Hausdorff metric. From now on, let X be the σ-field on CB generated by open sets with respect to the Hausdorff metric ρ; see [15]. A set-valued map C : Ω → CB is said to be measurable iff C is (F, X )-measurable. Such a map is often called a closed random set.
For a given closed random set B, the Ω-limit set of B is defined to be the set We denote by F u the σ−algebra of universally measurable sets associated to the measurable space (Ω, F). As far as we are aware, the following definition appeared for the first time as Definition 3.4 in the fundamental work by Flandoli and Schmalfuss [26], see also [8, Definition 2.6] (where however the first sentence below is missing). Definition 4.5. We say that an RDS ϑ-cocycle ϕ defined on a separable Banach space X is D-asymptotically compact iff for each D ∈ D, for every ω ∈ Ω, for any positive sequence (t n ) such that t n → ∞ and for any sequence {x n } such that

Definition 4.4. Let us assume that D is a family of random closed and bounded sets. A random set
x n ∈ D(ϑ −tn ω), for all n ∈ N, the set {ϕ(t n , ϑ −tn ω)x n : n ∈ N} is relatively compact in X.
Now we need to state a result on the existence of a random D-attractor, see Theorem 2.8 in [8] and references therein.  F, P, ϑ) is a metric DS, X is a Polish space, D is a nonempty class of closed and bounded random sets on X and ϕ is a continuous, D-asymptotically compact RDS on X (over T). Assume that there exists a D-absorbing closed and bounded random set B on X, i.e. for any which is F u -measurable. 3 Remark 4.7. If D contains every bounded and closed nonempty deterministic subsets of X, then as a consequence of our Theorem 4.6, [19, Theorem 2.1] and of [18,Corollary 5.8] we deduce that the random attractor A is given by where the union in equality (4.3) is taken over all bounded and closed nonempty deterministic subsets C of X.
Suppose that Assumption 3.4 is satisfied. If u s ∈ H, s ∈ R, f ∈ V and W t , t ∈ R is a two-sided Wiener process introduced after (3.5) such that the Cameron-Martin (or Reproducing Kernel Hilbert) space of the Gaussian measure L(w 1 ) is equal to K. A process u(t), t ≥ 0, with trajectories in C([s, ∞); ) is a solution to problem (3.19) iff u(s) = u s and for any φ ∈ V , t > s, (4.5) In the framework as above, suppose that u(t) = z α (t)+v α (t), t ≥ s, where v α is the unique solution to problem (3.21)- (3.22) with initial data u 0 − z α (s) at time s. If the process u(t), t ≥ s, has trajectories in C ([s, ∞); H) ∩ L 2 loc ([s, ∞); V ) ∩ L 2 loc ([s, ∞); L 4 (S 2 )), then it is a solution to problem (3.19). Vice-versa, if a process u(t), t ≥ s, with trajectories in C([s, ∞); ) is a solution to problem (3.19), then for any α ≥ 0, a process v α (t), t ≥ s, defined by z α (t) = u(t) − v α (t), t ≥ s, is a solution to (3.

21) on [s, ∞).
Our previous results yield the existence and the uniqueness of solutions to problem (3.19) as well as its continuous dependence on the data (in particular on the initial value u 0 and the force f ). Moreover, if we define, for x ∈ H, ω ∈ Ω, and t ≥ s, u(t, s; ω, u 0 ) := ϕ(t − s; ϑ s ω)u 0 = v(t, s; ω, u 0 − z(s)) + z(t), (4.6) then for each s ∈ R and each u 0 ∈ H, the process u(t), t ≥ s, is a solution to problem (3.19).
The following lemma is given in [9, Lemma 6.5], however the proof there is not quite correct. The bound on the nonlinear term b (v, z, v) there was not treated correctly, hence the power on the stochastic term z L 4 was not correctly stated. The error propagated to the rest of the paper. We present a corrected proof here.
By adding and subtracting ν λ1 2 v(t) 2 from (4.10) we find that Hence, multiplying the above by exp −νλ 1 t + 27C 4 This, together with Lemma 4.9 concludes the proof in the case of function r 2 . In the case of r 1 , we have Thus, by Lemma 4.9 we infer that r 1 also belongs to the class R. The argument in the case of function r 3 is similar but for the sake of the completeness we include it here. From the first part of the proof we infer that Since by Lemma 4.10, The proof in the other cases is analogous.
We have the following trivial results.
Proposition 4.13. The class R is closed with respect to sum, multiplication by a constant and if r ∈ R, 0 ≤r ≤ r, thenr ∈ R.
Now we are ready to state and prove the main result of this paper. A result of similar type for the Navier-Stokes equations on some 2-dimensional unbounded domain has been discussed in [8]. (iii) the family A of sets defined by A(ω) = Ω B (ω) for all ω ∈ Ω, is the minimal DR-attractor for ϕ, iŝ F-measurable, and (4.13) where the union in (4.13) is taken over all bounded and closed nonempty deterministic subsets C of H.
Proof. In view of Theorem 4.6 and Remark 4.7, it is enough to show (i) and (ii). The proof of (ii) will be done in the next proposition.
In other words, u(0, s; ω, x) ∈ B(ω), for all s ≥ t D (ω). This proves that B absorbs D. The proof of the proposition is adapted from [8], in which a RDS generated by NSEs on some 2dimensional unbounded domain was considered. The proposition generalises the asymptotically compactness of the RDS in [10,Proposition 8.1] to the DRasymptotically compactness of the RDS. Proof. Suppose that B is a closed random set from the class DR and K ∈ DR is a close random set which absorbs B. We fix ω ∈ Ω. Let us take an increasing sequence of positive numbers (t n ) ∞ n=1 such that t n → ∞ and an H-valued sequence (x n ) n such that x n ∈ B(ϑ −tn ω), for all n ∈ N.
Step I. Reduction. Since K(ω) absorbs B, for n ∈ N sufficiently large, ϕ(t n , ϑ −tn ω)B ⊂ K(ω). Since K(ω) is closed and bounded, and hence weakly compact, without loss of generality we may assume that ϕ(t n , ϑ −tn ω)B ⊂ K(ω) for all n ∈ N and, for some y 0 ∈ K(ω), ϕ(t n , ϑ −tn ω)x n → y 0 weakly in H. (4.18) Since z(0) ∈ H, we also have In particular, We claim that it is enough to prove that for some subsequence {n } ⊂ N Indeed, since H is a Hilbert space, (4.19) in conjunction with (4.20) imply that Therefore, in order to show that {ϕ(t n , ϑ −tn ω)x n } n is relatively compact in H we need to prove that (4.20) holds true.
Since K(ϑ −1 ω) absorbs B, there exists a constant N 1 (ω) ∈ N, such that Hence we can find a subsequence {n } ⊂ N and y −1 ∈ K(ϑ −1 ω) such that ϕ(−1 + t n , ϑ −t n ω)x n → y −1 weakly in H. (4.21) We observe that the cocycle property, with t = 1, s = t n − 1, and ω being replaced by ϑ −t n ω, reads as follows: Hence, by the last part of Theorem 3.13, from (4.18) and (4.21) we infer that ϕ(1, ϑ −1 ω)y −1 = y 0 . By induction, for each k = 1, 2, . . . , we can construct a subsequence As above, the cocycle property with t = k, s = t n (k) and ω being replaced by ϑ −t n (k) ω yields Hence, from (4.22) and by applying the last part of Theorem 3.13, we get where w-lim denotes the limit in the weak topology on H. The same proof yields a more general property: Before continuing with the proof, let us point out that (4.24) means precisely that y 0 = u(0, −k; ω, y −k ), where u is defined in (4.6).
Step III. Proof of (4.20). From now on, unless explicitly stated, we fix k ∈ N, and we will consider problem (3.19) on the time interval [−k, 0]. From (4.6) and (4.23), with t = 0 and s = −k, we have From (4.25) and (4.8) with t = 0 and τ = −k we infer that It is enough to find a nonnegative function h ∈ L 1 (−∞, 0) such that lim sup For, if we define the diagonal process (m j ) ∞ j=1 by m j = j (j) , j ∈ N, then for each k ∈ N, the sequence (m j ) ∞ j=k is a subsequence of the sequence (n (k) ) and hence by (4.27), lim sup j ϕ(t mj , ϑ −tm j ω)x mj − z(0) 2 ≤ −k −∞ h(s)ds + y 0 − z(0) 2 . Taking the k → ∞ limit in the last inequality we infer that which proves claim (4.20).
Step IV. Proof of (4.27). We begin with estimating the first term on the RHS of (4.26). If −t n (k) < −k, then by (4.6) and (4.8) we infer that First we will find a nonnegative function h ∈ L 1 (−∞, 0) such that lim sup This will be accomplished as soon as we prove the following four lemmas.
Proof of Lemma 4. 16. We recall that for α ∈ N, z(t) = z α (t), t ∈ R, being the Ornstein-Uhlenbeck process from Sect. 3.2, one has Let us recall that the spaceΩ(ξ, E) was constructed in such a way that Therefore, since the embedding X → L 4 (S 2 ) is a contraction, we have for n (k) sufficiently large, Since the set B is bounded in H, there exists ρ 1 > 0 such that for all n (k) , x n (k) ≤ ρ 1 . Hence lim sup Proof of Lemma 4.17. Because of (3.9), we can find ρ 2 ≥ 0 and s 0 < 0, such that, Hence by (4.30) we infer that lim sup This concludes the proof of Lemma 4.17.
Proof of Lemma 4.18.
we only need to show that the integrals It is enough to consider the case of z(s) 4 L 4 since the proof will be similar for the remaining case. Reasoning as in (4.30), we can find t 0 ≥ 0 such that for Taking into account the inequality (4.32), we have z(t) ≤ ρ 2 (1 + |t|), t ∈ R. Therefore, with ρ 3 := exp( By the continuity of all relevant functions, we can let t 0 → 0 to get the result. Therefore, the proof of (4.29) is concluded, and it only remains to finish the proof of (4.27). Let us denote by From (4.22) and the last part of Theorem 3.13 we infer that v n (k) → v k weakly in L 2 (−k, 0; V ). On the other hand, using the same methods as those in the proof of Theorem 3.12, there exists a subsequence of {v n (k) }, which, for the sake of simplicity of notation, is denoted as the old one which satisfies v n (k) → v k strongly in L 2 (−k, 0; L 2 loc (S 2 )).

Invariant Measure
In this section we consider the existence of an invariant measure. The main result in this section, i.e. Theorem 5.2 is a direct consequence of Corollary 4.4 [16] and Theorem 4.14 from the present paper about the existence of an attractor for the RDS generated by the stochastic Navier-Stokes equations (3.19). Let ϕ be the RDS corresponding to the SNSEs (3. 19) which has been defined in (4.4). We define the transition operator P t by a standard formula. For g ∈ B b (H), we put t, ω, x))] dP(ω), x ∈ H. (5.1) As in [10, Proposition 3.8] we have the following result whose proof is simply a repetition of the proof from [10] Proposition 5.1. The family (P t ) t≥0 is Feller, i.e. P t g ∈ C b (H) if g ∈ C b (H). Moreover, for any g ∈ C b (X), P t g(x) → g(x) as t 0.
Following [16] one can prove that ϕ is a Markov RDS, i.e. P t+s = P t P s for all t, s ≥ 0. Hence by [10, Corollary 3.10] which says that a time-continuous and continuous asymptotically compact, Markov RDS ϕ admits a Feller invariant probability measure μ, i.e. a Borel probability measure μ and P t (x, ·) is the transition probability, P t (x, Γ) = P t 1 Γ (x), x ∈ H. A Feller invariant probability measure for a Markov RDS ϕ on H is, by definition, an invariant probability measure for the semigroup (P t ) t≥0 defined by (5.1). Therefore, we obtain the following result. Remark 5.3. We believe that the uniqueness of an invariant measure for nondegenerate noise will follow from the classical procedure based on Doob's Theorem, see e.g. Seidler's paper [38] and references therein. If the noise is degenerate and spatially smooth, it seems that the results from a recent paper by Hairer and Mattingly [29] should be applicable in our setting. In particular, [29,Theorem 8.4], which gives a sufficient conditions for uniqueness in terms of controllability, should be applicable. Details will be published elsewhere. One should point out that these authors use the "vorticity" formulation and their initial data belongs to the L 2 space. This corresponds to our approach with the initial data belonging to the finite enstrophy space H 1 . However, we work in the space of finite energy, which seems to be physically more natural. On the other hand, verifying the sufficient conditions could be more challenging.
For the NSE without the Coriolis force this problems has been investigated in [1]. Corresponding NSE with the Coriolis force study is postponed till the next publication.
Acknowledgements. The authors would like to thank an anonymous referee for helpful comments to improve the presentation of the paper and to thank Gaurav Dhariwal for careful reading of the paper.

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