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A moderate deviation principle for 2D stochastic primitive equations

In this paper, we establish a central limit theorem and a moderate deviations for 2D stochastic primitive equations with multiplicative noise. The proof is mainly based on the weak convergence approach.


Introduction
As a fundamental model in meteorology, the primitive equations were derived from the Navier-Stokes equations, with rotation, coupled with thermodynamics and salinity diffusion-transport equations, by assuming two important simplifications: the Boussinesq approximation and the hydrostatic balance (see [21,22,26] and the references therein). This model in the deterministic case has been intensively investigated because of the interests stemmed from physics and mathematics. For example, the mathematical study of the primitive equations originated in a series of articles by Lions, Temam, and Wang in the early 1990s (see [21,22,23,24]), where they set up the mathematical framework and showed the global existence of weak solutions. Cao and Titi developed a beautiful approach to dealing with the L 6 -norm of the fluctuationṽ of horizontal velocity and obtained the global well-posedness for the 3D viscous primitive equations in [4].
For the primitive equations in random case, many results have been obtained. Debussche, Glatt-Holtz, Temam and Ziane established the global well-posedness of the strong solution of the primitive equations driven by multiplicative random noises in [5]. The ergodic theory of 3D stochastic primitive equations driven by regular multiplicative noise was studied in [8], where we proved that all weak solutions which are limits of spectral Galerkin approximations share the same invariant measure. Using a new method we proved the existence of random attractor for 3D stochastic primitive equations driven by fractional noise in [28]. In [15], Gao and Sun obtained a Freidlin-Wentzell's large deviation principle (LDP) for the stochastic primitive equations in two dimensional case while we established the same result in three dimensional case in [9].
In this paper, we shall investigate deviations of the strong solution Y ε (see (3.23))of 2D stochastic primitive equations from the deterministic solution Y 0 (see (3.24)), as ε decreases to 0, that is, the asymptotic behavior of the trajectory, where λ(ε) is some deviation scale which strongly influences the asymptotic behavior of Z ε . Concretely, three cases are involved.
(2) The case λ(ε) = 1 provides the central limit theorem (CLT). We will show that Z ε converges to a solution of a stochastic equation as ε decrease to 0 in Sect. 4.
(1.1) Similar to LDP, MDP arises in the theory of statistical inference naturally, which can provide us with the rate of convergence and a useful method for constructing asymptotic confidence intervals ( see, e.g. [11,12,19,20] and references therein). The proof of moderate deviations is mainly based on the weak convergence approach, which is developed by Dupuis and Ellis in [10]. The key idea is to prove some variational representation formula about the Laplace transform of bounded continuous functionals, which will lead to proving an equivalence between the Laplace principle and LDP. In particular, for Brownian functionals, an elegant variational representation formula has been established by Boué, Dupuis [2] and Budhiraja, Dupuis [3].
Up to now, there are some results about the moderate deviations for fluid dynamics models and other processes. For example, Wang, Zhai and Zhang [25] established the CLT and MDP for 2D Navier-Stokes equations with multiplicative Gaussian noise in the state space C([0, T ]; H) ∩ L 2 ([0, T ]; V). Further, the MDP for 2D Navier-Stokes equations driven by multiplicative Lévy noises were considered by Dong et al. [7] in state space D([0, T ]; H) ∩ L 2 ([0, T ]; V). In view of the characterization of the super-Brownian motion (SBM) and the Fleming-Viot process (FVP), Fatheddin and Xiong obtained the MDP for those processes in [13]. In this paper, we consider the CLT and MDP of 2D stochastic primitive equations, the state space is chosen to be C([0, T ]; H) ∩ L 2 ([0, T ]; V).
Compared with 2D Navier-Stokes equations, the 2D primitive equations are more complex because the nonlinear term of the 2D NSEs is of the form velocity × derivative of velocity, while the 2D viscous PEs has a challenging term: derivative of horizontal velocity × derivative of horizontal velocity which results in essential difficulties in obtaining moment estimates. An additional term |∂ z Y| 2 appears when estimating the nonlinear term B(Y, Y) in the functional space H. To overcome this difficulty, we take advantage of the special geometrical structure of 2D primitive equations to obtain estimate of |∂ z Y| 2 , which is also essential to establish some tightness results. With the help of those estimates, a central limits theorem and a moderate derivations principle for 2D primitive equations are proved.
This paper is organized as follows. The mathematical framework of 2D primitive equations is in Sect. 2. In Sect. 3, the formulation of 2D primitive equations: functional spaces, hypothesises on the noise and definition of solution are presented. The central limit theorem is proved in Sect. 4. We obtain the moderate deviation principle by using the weak convergence method in Sect. 5.

Preliminaries
Consider the 2D primitive equations driven by a stochastic forcing in a Cartesian system where the velocity v = v(t, x, z) ∈ R, the vertical velocity θ, the temperature T and the pressure P are all unknown functionals. (x, z) ∈ M = (0, L) × (−h, 0), t > 0, W 1 and W 2 are two independent cylindrical Wiener processes, which will be given in Sect. 3. ∆ = ∂ 2 x + ∂ 2 z is the Laplacian operator. Without loss of generality, we assume that µ 1 = µ 2 = 1.
We impose the following boundary conditions: Integrating (2.4) from −h to z and using (2.6)-(2.7), we have Then, (2.2)-(2.5) can be rewritten as The boundary value conditions for (2.10)-(2.12) are given by Denote Y = (v, T ) and the initial condition 3 Formulation of the SPDE

Some Functional Spaces
Let L(K 1 ; K 2 ) (resp. L 2 (K 1 ; K 2 )) be the space of bounded (resp. Hilbert-Schmidt) linear operators from the Hilbert space K 1 to K 2 , whose norm is denoted by · L(K 1 ;K 2 ) ( · L 2 (K 1 ;K 2 ) ). For p ∈ Z + , set In particular, | · | and (·, ·) represent norm and inner product of L 2 (M) (or L 2 ((0, L))), respectively. For m ∈ N + , (W m,p (M), · m,p ) stands for the classical Sobolev space, see [1]. When p = 2, we denote by Taking into account the boundary conditions (2.13)-(2.15), the inner product and norm on V can be given by Note that under the above definition, a Poincaré inequality |Y| ≤ C Y holds for all Y ∈ V. LetV be the closure of V ∩ (C ∞ (M)) 2 in (H 2 (M)) 2 and equip this space with the norm and inner product of H 2 (M).
Define the intermediate space Let V ′ be the dual space of V. Then, the dense and continuous embeddings hold and denote by x, y the duality between x ∈ V and y ∈ V ′ .

Some Functionals
Define P H be the Leray type projection operator from (L 2 (M)) 2 onto H. The principle linear portion of the equation is defined by where D(A) is the closure ofH with respect to the topology of H 2 (M).
It's well-known that A is a self-adjoint and positive definite operator. Due to the regularity results of the Stokes problem of geophysical fluid dynamics, we have |AY| |Y| H 2 (O) ( see [29]).
By interpolation inequalities (see [15,16], etc), we have For the pressure term in (2.10), define Using the above functionals, we obtain

Definition of Strong Solution
For the strong solution of (3.20), we shall fix a single stochastic basis T := (Ω, F , {F t } t≥0 , P, W) with the expectation E. Here, In order to obtain the global well-posedness and moderate deviations of (3.20), we introduce the following Hypothesises: Hypothesis B For the same constant K, we suppose that ∂ z ψ(t, φ) 2 Now, we give the definition of strong solution in probability to (3.20).
, P, W) be a fixed stochastic basis and the initial condition Y 0 ∈ H.
To study the long-time behavior of the system (3.20), some kinds of V−norm estimates are needed, but it is difficult for our model. Fortunately, taking advantage of the special geometry structure of (3.20), we only need the moment estimate of ∂ z v in H, which has been obtained during the proof of global well-posedness of the stochastic system (see [15,16]).
As the parameter ε tends to 0, the solution Y ε of (3.23) will tend to the solution of the following SPDE As stated in the introduction, we will investigate the asymptotic behavior of the trajectory, where λ(ε) is equal to 1 or satisfies (1.1).

Central limit theorem
In this part, we will estimate the central limit theorem, i.e. λ(ε) = 1 in (3.25). Let Y ε be the unique solution of (3.23) in L 2 (Ω; C([0, T ]; H)) ∩ L 2 (Ω; L 2 ([0, T ]; V)) and Y 0 be the unique solution of (3.24). Taking an similar argument as in the proof of Theorem 3.1, we can also have Lemma 4.1. Let Y 0 ∈ H. Assume that Hypothesis A and Hypothesis B hold, then there exists ε 0 > 0 such that In particular, it holds that where C(K, T ) is a constant that depends only on K and T . Now, we explore the convergence of Y ε as ε → 0.
Applying Itô formula to |X ε | 2 and by (3.18), we have (4.32) Taking the supremum up to time t ∧ τ N in (4.32), and then taking the expectation, we have By (3.19) and the Young's inequality, we have By the Cauchy-Schwarz inequality and the Young's inequality, we obtain Moreover, by Hypothesis A, we deduce that Applying the Burkholder-Davis-Gundy inequality and Hypothesis A, we have Combing the above estimates and (4.26), we get By Gronwall inequality, (4.28) and (4.29), we have LetṼ 0 = (ṽ 0 ,T 0 ) be the solution of the following SPDE: withṼ 0 (0) = 0. Under Hypothesis A and Hypothesis B, the global existence and uniqueness of the strong solution to (4.35) can be proved easily by changing (4.35) into a linear partial differential equation with random coefficients. Taking a similar argument as in Lemma 4.1, we have The following is the first result in this article: central limit theorem.
From (4.35) and (4.39), we have Applying Itô formula to |ρ ε (t)| 2 and by (3.18), we have Taking the supremum up to time t ∧ τ N in the above equation, and taking the expectation, we obtain With the help of (3.18), (3.19), integration by parts and the Young's inequality, we have By (3.19), we get By Cauchy-Schwarz inequality and the Young's inequality, we otain Applying the Burkholder-Davis-Gundy inequality to J 4 (t) and by Hypothesis A, we obtain By Hypothesis A, we have Collecting the above estimates and by (4.26), (4.28), we have Suppose that there exists a constant ε 0 > 0 such that which will be proved in the following Lemma 4.2. Then, Applying Gronwall inequality to (4.41) and by (4.28), (4.29), we obtain Letting N → ∞, we conclude the result.

Moderate deviation principle
In this part, we are concerned with the moderate deviation principle of Y ε . As stated in introduction, we need to prove Y ε −Y 0

The weak convergence approach
√ ελ(ε) , we will use the weak convergence approach introduced by Budhiraja and Dupuis in [3] to verify Z ε satisfies a large deviation principle. Firstly recall some standard definitions and results from the large deviation theory (see [6]).
Suppose {Z ε } be a family of random variables defined on a probability space (Ω, F , P) taking values in some Polish space E.
where A o andĀ denote the interior and closure of A in E, respectively.
(ii) (Laplace principle) The sequence {Z ε } is said to satisfy the Laplace principle with rate function I if It's well-known that the large deviation principle and the Laplace principle are equivalent if E is a Polish space and the rate function is good (see [6]).
Suppose W(t) is a cylindrical Wiener process on a Hilbert space U defined on a probability space (Ω, F , {F t } t∈[0,T ] , P) ( the paths of W take values in C([0, T ]; U), where U is another Hilbert space such that the embedding U ⊂ U is Hilbert-Schmidt). Now we define Here, we use the weak topology on L 2 ([0, T ]; U) under which T M is a compact space.
Suppose G ε : C([0, T ]; U) → E is a measurable map and Z ε = G ε (W). Now, we list the following sufficient conditions for the Laplace principle (equivalently, large deviation principle) of Z ε .
Hypothesis H1 There exists a measurable map  The following result is due to Budhiraja et al. in [3].
Theorem 5.1. If G 0 satisfies Hypothesis H1, then Z ε satisfies a large deviation principle on E with the good rate function I given by By convention, I(∅) = ∞.
From (4.39), Z ε satisfies the following SPDE The following is our main result in this part.

Compactness of R h
In order to prove compactness of R h , as in [14], we introduce the following space. Let K be a separable Hilbert space. The following result can be found in [14]. The following is the result we obtain in this part.  With the aid of the Hölder inequality and the Cauchy-Schwarz inequality, we have By the Hölder inequality, we obtain From the definition of W r,2 ([0, T ]; V ′ ), for r ∈ (0, 1 2 ), we have Thus, by (5.52) and (5.53), we have E I 1 W r,2 ([0,T ];V ′ ) ≤ C 1 (r, M, K, T ).

Tightness ofZ ε
For any h ε ∈ A M , consider In view of properties of Z ε and R h , by the same method as Theorem 5.3, we can obtain Proof. Define From (5.55), we havē Let α > 1 be fixed, we deduce from the interpolation inequality that D(A α 2 ) ⊂ L ∞ (M). Then, with the aid of the Hölder inequality and the Cauchy-Schwarz inequality, we have By the Hölder inequality, we obtain From the definition of W r,2 ([0, T ]; D(A − α 2 )), for r ∈ (0, 1 2 ), we have Thus, by (4.28) and (4.29), we have Referring to (4.28), (5.56) and by Hypothesis A, for any r ∈ (0, 1 2 ) and p ≥ 2, we have ≤ C 2 (r, K, T ).

Proof of Theorem 5.2
According to Theorem 5.1, the proof of Theorem 5.2 will be completed if the following Theorem 5.7 and Theorem 5.8 are established. (ii) R h n → R h in L 2 ([0, T ]; D(A)) weakly.