Exponential mixing for the fractional Magneto-Hydrodynamic equations with degenerate stochastic forcing

We establish the existence, uniqueness and exponential attraction properties of an invariant measure for the MHD equations with degenerate stochastic forcing acting only in the magnetic equation. The central challenge is to establish time asymptotic smoothing properties of the associated Markovian semigroup corresponding to this system. Towards this aim we take full advantage of the characteristics of the advective structure to discover a novel H\"ormander-type condition which only allows for several noises in the magnetic direction.


Introduction
The dynamics of the velocity and the magnetic field in electrically conducting fluids and basic physics conservation laws can be described by the Magneto-Hydrodynamic(MHD) equations (c.f. [3,4]). The existence, uniqueness, regularity and stability of the MHD equations have been extensive studied in many papers, see [6,7,19,24].
Meanwhile, for the MHD equations driven by non-degenerate stochastic forcing terms both in the velocity and in the magnetic field, the existence and uniqueness of invariant measure was obtained via coupling method in [2]. Huang and Shen [12] proved the well-posedness and the existence of a random attractor for the stochastic 2D incompressible fractional MHD equations driven by Gaussian multiplicative noise. For the stochastic fractional MHD equations with degenerate multiplicative noise on the Torus T 2 , Shen, Huang and Zeng [26] proved the existence and uniqueness of the invariant measure for the associated transition semigroup. The noise in [26] is degenerate in the sense that it drives the system only in the first finite Fourier modes.
Hypothesis 1.1 may not seem intuitive, however it falls into the most interesting case of degenerate noise-"hypoellipticity" setting, and includes many interesting examples, see Remark 2.1 and Example 2.1. Now we state our main result as below.
Theorem 1.1. Assume that Hypothesis 1.1 holds, then the associated Markov semigroup corresponding to 1.2 possesses a unique, exponentially mixing invariant measure. Furthermore, a law of large numbers together with a central limit theorem is established under the current circumstances.
We remark that Theorem 1.1 is a simplified version of Theorem 2.1 and refer readers to Section 2.2 for more details.
Nowadays ergodicity research on infinite-dimensional systems driven by degenerate stochastic forcing has attracted considerable attention ( [1,8,9,10,11,13,15,16,17,20,22]), not only because this poses many interesting mathematical challenges, but also provides rigorous justification for the explicit or implicit statistical measurement assumptions invoked in a physical environment. It is exciting that recently there have been remarkable breakthroughs (c.f. [9,10,13]), initiating the development a theory of "hypoellipticity" for degenerated forced infinite-dimensional stochastic systems. However, the whole theory is far from mature and remains in an involved formation. The reason for this is unlike in the case of finite-dimensional systems, the invertibility of Malliavin matrix is hard to prove, not to mention characterise its range. Experts have thus devised a tactful strategy to take full advantage of the structure of turbulent systems. Roughly speaking, infinite-dimensional as these systems are, their unstable directions are confined to be finitely many, and it is reasonable that one just focus on proving the Malliavin matrix to possess small eigenvalues on some spanning cones.
The technical difficulties of this method lie in how to generate successively larger finite dimensional spaces through the interaction between the nonlinear and stochastic terms and how to exert delicate spectral analysis on these spaces. To be more specific, one digging into the technical details will find that the proof virtually relies heavily on the results of progressive computation of Lie brackets using constant vector fields and nonlinear terms, by virtue of which the whole involved arguing process will be decomposed in an inductive manner and most importantly an appropriate Hörmander condition will thus be determined. It is worth emphasizing that finding out such an ideal collection of Lie brackets to accomplish the task is case-by-case, there is no general recipe for all. For instance, Navier-Stokes equations and Boussinesq equations are treated quite differently and therefore lead to different Hörmander condition (c.f. [9,10,13]).
The main contribution of the manuscript is, we successfully devised a special pattern of Lie bracket computations suitable for the fractional MHD equation, and thus propose a novel Hörmander condition. Apart from [9,10], the considered fractional MHD equations are of original formation instead of vorticity formation. Furthermore, Due to the special form of stochastic fractional Magneto-Hydrodynamic equations (1.2), we exert a series of Lie bracket computation strategically to exploit the distinctive structure of nonlinear advective terms. Roughly speaking, we activate the noise term within the magnetic equation to spread to the velocity equation through advection, then perturb it again with stochastic forcing to generate new directions in the b component of the phase space. On the flip side, new v directions can be generated similarly except being stochastically driven once. This procedure can be repeated iteratively so as to span the whole phase space as long as Hypothesis 1.1 is satisfied (c.f. Section 4 for more details). Attentive readers may find that the whole deductive process and derived Hörmander condition distinguish from that within [9,10,13].
We would also like to add that the degenerate noise in [25,26] is in a fairly simple manner and belongs to the so-called "essentially elliptic" setting. More precisely, although driven modes are assumed to be finite, they are forced to be one by one and required to be sufficiently many, while in this paper we adopt a hypoellipticity setting, which allows for a limit number of directions to be driven on and off. We will further exemplify this essential difference with Example 2.1, which also exhibits a distinct picture compared with [9,10,13]. All in all, our analysis gets the utmost out of existing techniques in the recent works but yields something peculiar and we believe it will enrich ergodic research upon systems of SPDEs.
This article is organized as follows: In Section 2 we introduce general definitions and formulate our main result (Theorem 2.1). Section 3 is devoted to some moment estimates which will be used frequently. In Section 4 we illustrate progressive computations of Lie brackets in detail. Then in Section 5 we focus on proving the spectral properties of Malliavin matrix (Theorem 5.1) and give a gradient estimate of the Markov semigroup (Proposition 5.3). Finally, we provide a proof of Theorem 2.1 in Section 6.

Preliminaries
2.1. Mathematical setting. In this section we introduce a functional setting for the equations (1.2). Then we describe specifically the stochastic forcing, and thus formulate (1.2) as an abstract stochastic evolution equation. Finally, we introduce some basic elements of the Malliavin calculus centering on the Malliavin matrix.
The higher order Sobolev spaces are denoted by H s 1 := u = (u 1 , u 2 ) ∈ (W s,2 (T 2 )) 2 : ∇ · u = 0, We also denote H −s := (H s ) * the dual space to H s . Specially, The norm on the space H 1 is given by Likewise, let H := H 0 . By a slight abuse of notation, ·, · may denote the inner product on Hilbert space H or H 1 . Let Π be the projection operator from (L 2 (T 2 )) 2 to the space H 1 .
For any m, n ∈ R, we denote by We also denote H −m;−n := (H m;n ) * the dual space to H m;n . Next, we need to construct the stochastic forcing based on an orthogonal basis of H, therefore for k = (k 1 , k 2 ), denote It is commonsense that {e m k } k∈Z 2 \{0,0},m∈{0,1} forms an orthogonal basis of H 1 exactly. Denote Denote the Hilbert-Schmidt norm of Q b by We consider stochastic forcing of the form With these preliminaries in hand, the equations (1.2) may be written as an abstract stochastic evolution equation or in a more compact formulation We say that U = U(t, U 0 ) is a solution of (2.14) if it is F t -adapted, U ∈ C([0, ∞); H) ∩ L 2 loc ([0, ∞); H 1 ) a.s. and U satisfies (2.3) in the mild sense, that is, The well-posedness can be established similarly as in [12]. Hence we let U = U(t, U 0 ) be the unique solution of (2.3) with initial value U 0 . For any ξ = (ξ 1 , ξ 2 ) ∈ H, t s 0, the Jacobian J s,t ξ is actually the unique solution of In the interest of brevity, set J t ξ := J 0,t ξ. Let J (2) s,t : H → L(H, L(H)) be the second derivative of U with respect to an initial value U 0 . Observe that for fixed U 0 ∈ H and any ξ, ξ ′ ∈ H the function ̺ = ̺ t := J (2) s,t (ξ, ξ ′ ) is the solution of where ∇B(θ)ϑ = B(θ, ϑ) + B(ϑ, θ).
One may infer from Duhamel's formula that (c.f. [13]) for v ∈ L 2 (Ω; L 2 (0, T ; R d )), We define the random operator A s,t : Direct computation shows that A s,t v satisfies the following equation For any s < t, let A * s,t : H → L 2 (s, t; R d ) be the adjoint of A s,t , then (A * s,t ξ)(r) = Q * b K r,t ξ, for any ξ ∈ H, r ∈ [s, t], where Q * b : H → R d is the adjoint of Q b , and for s < t, K s,t ξ is the solution of the following "backward" system It is time to define the Malliavin matrix as This equation enables us to translate the ergodicity issue into a control problem. Actually in conjunction with the Malliavin integration by parts formula, one can obtain the estimate on ∇P t Φ through spectral analysis on the Malliavin matrix M (c.f. Section 5).

Main theorem.
Before stating the main theorem of the manuscript, let us recall some basic notations with regard to the associated Markovian semigroup. It is necessary to introduce new functional spaces first. Denote by M b (H) and C b (H) respectively, the spaces of bounded measurable and bounded continuous real valued functions on H equipped with the supremum norm. We also define for any η > 0, which is the special admissible functional space for Theorem 2.1.
The transition function associated to (2.3) is given by where B(H) is the collection of Borel sets on H, U(t, U 0 ) is the solution of (2.3) with initial value U 0 . We also define the Markov semigroup {P t } t 0 with P t : Now we will give our main results in this article.
Theorem 2.1. Assume Hypothesis 1.1 holds, then there exists an unique invariant measure µ * associated to (1.2) and for each t 0 the map P t is ergodic relative to µ * . Moreover, there exists a constant η * such that µ * satisfies for each η ∈ (0, η * ) holds for any Φ ∈ O η , U 0 ∈ H and any t 0. (ii) (Weak law of large numbers) For any Φ ∈ O η and any U 0 ∈ H, (2.10) (iii) (Central limit theorem) For any Φ ∈ O η , every U 0 ∈ H and ξ ∈ R

11)
where X is the distribution function of a normal random variable with zero mean and variance equal to Remark 2.1. Interestingly, under Hypothesis 1.1 it is possible that the noise allows to be so degenerate that only four modes in the magnetic direction are actually driven. The next example allows one to get a primary idea into this phenomenon, and meanwhile to notice the specificity of Hypothesis 1.1 in comparison to [9,10,11].
Proof. For n 0, defineẐ then it is not difficult to check thatẐ Therefore, one sees that which yields the desired result.

Moment estimates on
In this section we provide moment bounds with respect to the unique solution U and its linearizations. They may seem familiar for readers who are familiar with research works with regard to ergodicity on the stochastic Navier-Stokes equations and so on. Hence some proofs are sketched or omitted if they do not distinguish from existing methods. However for the fractional MHD equations (1.2), we have to impose α > 1, β > 1 to compensate for the complicated advective operator B. This is accomplished through delicate interpolation and weighting. Lemma 3.1 and Lemma 3.2 give one a glimpse of this strategy.
(4) The proof of (4) follows similarly as in [18,Proposition 2.4.12] and the fact W k,ℓ has finite pth moment for any p 1.
The next lemmata include necessary estimates on linearizations of (2.3). Referring back to (2.5), (2.7) and (2.6), one finds that J s,t is the Jacobian operator with its adjoint K s,t and derivative J (2) s,t . At first glance the following bounds are closely related to those on the Malliavin derivative M s,t . The technically oriented readers may jump to Section 5 for further details.
Proof. Since ρ * For any N 1, define along with the associated projection operators The following three lemmas are particularly useful in translating the bounds on the Malliavin matrix into gradient estimates on the Markov semigroup (c.f. Proposition 5.3). Since their proofs adopt similar approach as above in combination with a straightforward modification of existing methods (c.f. [9,10]), they are omitted to save space. Lemma 3.4. For every p 1, T > 0, δ, γ > 0 there exists N * = N * (p, T, δ, γ), such that for any N N * one has Here, L(X, Y) denotes the operator norm of the linear map between the given Hilbert spaces X and Y. 1, Recall that D is the Malliavin derivative. We adopt the notions Then observe that for τ t Lemma 3.6. For any η > 0, ξ ∈ H and p 1 we have the bounds where C = C(η, p).

Details of Lie bracket computations
For any Fréchet differentiable E 1 , E 2 : H → H, [E 1 , E 2 ] is referred to as the Lie bracket of two "vector fields" E 1 , E 2 . This section is technical, however, reveals some intrinsic thoughts of the manuscript. As a matter of fact, we present that for any N ∈ N, how finite dimensional subspaces H N of H can be generated through the iterations of Lie brackets. It is worth mentioning that these computations are motivated by the celebrated Hörmander condition for the Kolmogorov-Fokker-Planck equations associated to (2.3). The following is split into two parts. Firstly, we describe how the velocity direction u is covered.
In fact, Y m k (U) and J m,m ′ k,ℓ (U) are devised elaborately by calculation to guarantee that the following two lemmas hold.

6)
where c is an absolutely non-zero constant independent of k, ℓ and may change from line to line.
Proof. Since all of the above can be proved in a similar way by direct calculating, we only give the proof of (4.3). It is from (4.2) that where ·, · denotes the inner product on H 1 , c is a non-zero constant.
By lemma 4.2 and (4.1), we can generate suitable directions in the u component.

Lemma 4.3.
Let a = k,ℓ ⊥ |k||ℓ| , then for some absolutely non-zero constant c which is independent of k, ℓ, the following inequalities hold.

4.2.
Covering magnetic direction. Likewise, we will need the following notations for the b direction, which are also obtained through the iteration of Lie brackets computation.
The following lemma is the counterpart of Lemma 4.2.
Lemma 4.4. Denote a = k,ℓ ⊥ |k||ℓ| , then for some absolutely non-zero constant c which is independent of k, ℓ(It may changes from line to line), the following equalities hold.
Proof. By the definition of Z m,m ′ k,ℓ , we get With a similar way, one sees that Combining (4.10)(4.9) with (4.8), we obtain Therefore, The proof of other equalities are similar.
Likewise, by Lemma 4.4 and (4.7) we can generate suitable directions in the b component.
Lemma 4.5. Denote a = k,ℓ ⊥ |k||ℓ| , then for some absolutely non-zero constant c which is independent of k, ℓ(It may changes from line to line), the following equalities hold.
In conclusion, we give an illustration in Figure 4.1 how the new directions generated from the existing directions via the iterations of the chain of bracket computations. The construction is interesting that in the upper half part ψ's are generated by σ's, while in the lower half part σ's are generated by ψ's. This antisymmetric relationship is originated from the advective structure in B. arrows with red color shows that the process is iterative. The doubled arrow with yellow color (⇒) shows that k ± ℓ is a element belongs to Z 2n+1 or Z 2n+2 actually.

Spectral properties of M
For any α > 0, N ∈ N, we define The aim of this section is to prove the following theorem, which gives information on the probability of eigenvectors with sizable projections in the unstable directions to have small eigenvalues. Broadly speaking, this provides us the invertibility of the Malliavin matrix on the space spanned by the unstable directions. Since it is finite dimensional under current circumstances, one can thus formulate a control problem through the Malliavin integration by parts formula to obtain the gradient estimate on the Markov semigroup, which is extremely useful in establishing ergodicity (c.f. Proposition 5.3).
Theorem 5.1. For any N 1, α ∈ (0, 1] and η > 0, there exists a positive constant ε * = ε * (α, η, N, T ) > 0, such that, for any n 0, and ε ∈ (0, ε * ], there exists a measurable set Ω ε = Ω ε (α, N, T ) ⊆ Ω satisfying where r = r(α, η, N, T ) : (0, ε * ] → (0, ∞) is a non-negative, decreasing function with lim ε→0 r(ε) = 0, and on the set Ω ε , In order to prove this theorem, we will first introduce a series of quadratic forms Q N and their lower bounds, next in Subsection 5.1 we introduce or recall some notational conventions and technical tools which will be used frequently. Then we estimate upper bounds on Q N in Subsection 5.2. Finally in Subsection 5.3, we complete the proof of Theorem 5.1. To start with, denote Lower bounds on these Quadratic forms are fairly simple since we are merely focusing on φ ∈ S α,N .
Proof. Its proof is trivial.
and by expanding U =Ū + Q b W we find where for any function g : [T/2, T ] → R, g C α is defined by and for α = 0, g C 0 is defined by Proof. (5.6) and (5.8) follow directly from Lemma 3.1 and Lemma 3.2. By the expressions of Y m k (Ū), F(U), Z m,m ′ k,ℓ and Lemma 3.1, there exist C = C(k, p, η) and q = q(p, k) such that This along with Lemma 3.2 yields Combining (5.10) with Lemma 3.2, one arrives that Immediately (5.7) follows from (5.11) and (5.12). The proof of (5.9) is similar to that of (5.7).

5.2.
Quadratic forms:upper bounds. The purpose of this subsection is to give a proof of the following proposition.
Roughly speaking, this theorem suggests that the quadratic forms Q N are bound to have small eigenvalues on S α,N with large probability once the Malliavin matrix M 0,T possesses a small eigenvalue.
Motivated by Section 4, we will adopt an iterative and inductive strategy to prove Theorem 5.2. To make this more precise, notice that Therefore we start from that M 0,T φ, φ is small to deduce that σ m ′ ℓ , K r,T φ are small, which is the content of Lemma 5.3. Then by Lie brackets computation as suggested by Figure 4.1, we estimate progressively that , K r,T φ and ψ m k+ℓ , K r,T φ are all small, which are the contents of Lemma 5.4, Lemma 5.5 and Lemma 5.6 respectively. We also need to integrate all these results, since they only hold on different large sets, which is the content of Lemma 5.10. Likewise, in the other direction we start from ψ m k , K r,T φ are small to estimate progressively that Y m k (U), K r,T φ , [Y m k (U), σ m ′ ℓ ], K r,T φ and σ m k+ℓ , K r,T φ are all small on some large sets, which are the contents of Lemma 5.7, Lemma 5.8 and Lemma 5.9 respectively. Lemma 5.11 serves to integrate all these results. The whole process is iterative and inductive so that we can tackle with successively larger finite dimensional subspace. To be specific, we refer the readers to Figure 5.1 for an illustration of the arguing structure in this subsection that lead to the proof of Proposition 5.2.
Proof. It directly follows from Lemma 4.3 and (4.1).

5.3.
Proof of Theorem 5.1. Now we are in a position to prove Theorem 5.1.
Once Proposition 5.2 is established, one can translate spectral bounds on the Malliavin matrix M to the estimate on ∇P t Φ. This constitutes the main content of the next proposition, and since the Malliavin matrix M only prove to be nondegenerate on finite dimensional cones, gradient estimates are bound to be in a asymptotic form, which leads Hairer [10] to introduce the celebrated conception of asymptotic strong Feller. Proposition 5.3. For some γ 0 > 0 and every η > 0, U 0 ∈ H, the Markov semigroup {P t } t 0 defined by (2.8) satisfies the following estimate ∇P t Φ(U 0 ) C exp (η U 0 2 ) P t (|Φ| 2 )(U 0 ) + e −γ 0 t P t ( ∇Φ 2 )(U 0 ) for every t 0 and Φ ∈ C b (H), where C = C(η, γ 0 ) is independent of t and Φ.
Proof. Ever since [10] the proof of this type of gradient inequality have been attached great importance to and improved all along. Now the method to prove it is more or less standard. Broadly speaking, supplied with moment estimates of U, J s,t ξ, K s,t ξ, J (2) s,t (ξ, ξ ′ ) listed in Section 2, one need to formulate a control problem through the Malliavin integration by parts formula, then do some decay estimates adopting an iterative construction with the aid of Lemma 3.4, Lemma 3.5, Lemma 3.6. We refer the readers to [9,10,11,13] and omit the details.
Proof of Theorem 2.1. (a) Ito's formula yields that U t 2 − U 0 2 + 2 for some constant C dependent on η. From this, Applying [11,Theorem 3.4] yields that the unique invariant measure µ * is exponentially mixing.
(e) To apply [14, Theorem 2.1], the following inequality is critical [ρ(0, U)] 3 P t (U 0 , dU) C exp (η * U 0 2 ), (6.4) where the constant C is independent of U 0 and t 0. With the definition of ρ, this can be easily established from Lemma 3.1. The proof is finished.