INVARIANT MEASURE OF STOCHASTIC FRACTIONAL BURGERS EQUATION WITH DEGENERATE NOISE ON A BOUNDED INTERVAL

. This work is concerned with the invariant measure of a stochastic fractional Burgers equation with degenerate noise on one dimensional bounded domain. Due to the disturbance and inﬂuence of the fractional Laplacian operator on a bounded interval interacting with the degenerate noise, the study of the system becomes more complicated. In order to get over the diﬃculties caused by the fractional Laplacian operator, the usual Hilbert space does not ﬁt the system, we introduce an appropriate weighted space to study it. Mean-while, we apply the asymptotically strong Feller property instead of the usually strong Feller property to overcome the trouble caused by the degenerate noise, the corresponding Malliavin operator is not invertible. We ﬁnally derive the uniqueness of the invariant measure which further implies the ergodicity of the stochastic system.

1. Introduction. Burgers equation plays an important role in describing the interaction of dissipative and non-linear inertial terms in the motion of the turbulent fluid [1]. Through the Hopf-Cole transformation [4,17], the interaction can be solved in an explicit expression, and the solution develops shock waves in the limit of vanishing viscosity. However, it is no longer available in most case of random forces.
In this paper, we consider the stochastic fractional Burgers equation with degenerate noise on one dimensional bounded domain (also called SFBE for short)   where D = (−1, 1) ⊆ R 1 , D c = R 1 \D, andẆ (t) is a degenerate noise specified in the next section in detail. The fractional Laplacian operator (−∆) s is defined as where C s is a constant depending on s.
Notice that the fractional Laplacian operator (−∆) s is defined on a bounded interval D, which is a nonlocal operator and is heavily different from the usual Laplacian operator −∆. Kwaśnicki [18] obtained the eigenvalues of the fractional Laplacian operator on a bounded interval. Du et al. [8] analyzed the characteristics of the fractional Laplacian operator. When the definition of the fractional Laplacian operator in (1.2) is extended to the whole space R 1 , it coincides with the definition by Fourier transform.
For the Burgers equation with the usual Laplacian operator driven by the normal noise, Bertini et al. [3] proved the existence of solutions by an adoption of the Hopf-Cole transformation. Da Prato et al. [6] also used semigroup method to establish the well-posedness of this system. E et al. [11] proved that there exists a unique stationary distribution. Gourcy [13] proved the large deviations principle. Goldys et al. [12] studied that strong Feller property and irreducibility to obtain the ergodicity of the system.
For the Burgers equation with the fractional Laplacian operator driven by the normal noise, Lv and Duan [19] established the existence of the martingale solution and the weak solution. Brzezniak, Debbi and Goldys [2] further studied the ergodicity properties of the system.
However, there are few works of the Burgers equation with the fractional Laplacian operator on bounded domains driven by the degenerate noise. In this paper, we are especially interested in the ergodicity of SFBE (1). Due to the disturbance and influence of the fractional Laplacian operator on a bounded interval interacting with the degenerate noise, the study of SFBE (1) becomes more complicated. There are three key points to achieve our goal: the first is to get over the difficulties, the usual Hilbert space does not fit the system, caused by the fractional Laplacian operator; the second is to overcome the trouble, the corresponding Malliavin operator is not invertible, caused by the degenerate noise; and the third is to deal with the nonlinearity in the case of interacting of the fractional Laplacian operator with the degenerate noise.
For the first key point, since that the fractional Laplacian operator defined on a bounded domain is nonlocal, the usual fractional Sobolev space does not fit the system. Combining with the fractional operator theory of Du et al. [8], we introduce an appropriate weighted function to construct weighted spaces to study it. For the second key point, SFBE (1) is driven by degenerate noise, which causes the corresponding Malliavin operator is not invertible, such that the usually strong Feller property is invalidating. We adopt the argument of Hairer and Mattingly [14], asymptotically strong Feller property, to solve it. For the third key point, perturbating by the interaction of degenerate noise and fractional Laplacian operator on the bounded interval, the estimate of the nonlinear term uu x of SFBE (1) becomes more complicated. We manage to use some subtle inequalities and techniques to conquer the troubles.
The rest of this paper is organized as follows. In the next section, we state the definition of classical fractional Sobolev spaces and define nonlocal weighted Sobolev spaces, then specify the degenerate white noise. In the last section, we prove the asymptotically strong Feller property and irreducibility which further implies the ergodicity of SFBE(1).

2.
Preliminaries. In this paper, let L 2 (D) be the usual Sobolev space, whose inner product and norm are ·, · and · , respectively. Further, let {e i (x)} i≥1 be an orthonormal basis of L 2 (D). Define L 2 l := span{e 1 , · · · , e N }, the projector π l : L 2 (D) → L 2 l , and π h := I − π l , where I is the identity operator. Then Similarly, for a complete probability space (Ω, F , P) and a Banach space X, the function space L r (Ω; X) can also be defined for r with 1 ≤ r < +∞.
(ii). For 0 < s < 1, let D be an open set in R 1 of class C 0,1 with bounded boundary and u be a measurable function from D to R 1 . Then, the following inequality holds u W s,2 (D) ≤ C u W 1,2 (D) for some suitable positive constant C. In particular, W 1,2 (D) ⊆ W s,2 (D).
for any k ∈ 2, 2 1−2s ; i.e. the space W s,2 (D) is continuously embedded in L k (D) for any k ∈ 2, 2 1−2s . If, in addition, D is bounded, then the space W s,2 (D) is continuously embedded in L k (D) for any k ∈ 2, 2 1−2s . (ii). For 0 < s < 1, let D ⊂ R 1 be an extension domain for W s,2 (D) with no external cusps. Then there exists a positive constant C = C(s, D), such that be the homogeneous Sobolev space and assume that p, p 0 , Weighted fractional Sobolev space. For s ∈ (0, 1), from Du et al. [8,9], we define the nonlocal divergence operator

and the adjoint operator
Here the vector mapping V(x, y), β(x, y) : |x−y| 1+2s and Θ be an identity matrix, then Furthermore, for D = (−1, 1) and u = 0 in D c , we deduce that Put ρ(x) := D c 2 |y−x| 1+2s dy, which is in (0, 4) for x ∈ D. Hence, we introduce a weighted fractional Sobolev space It is obvious that When 1 < s < 2, let s = 1 + σ with σ ∈ (0, 1). We can also define 2.3. Degenerate white noise. Let L 0,0 2 denote the space of Hilbert-Schmidt operators from L 2 (D) to L 2 (D), endowed with the norm φ 2 Let W (t) be a Wiener process, which is defined on a complete probability space (Ω,F , P) with the filtration {F t } t≥0 and takes value in the separable Hilbert space L 2 (D) with zero value on D c . Its covariance operator Q is a symmetric nonnegative operator satisfying T rQ < +∞. Furthermore, it satisfies the expansion as follows where N is a finite positive integer, the operator φ is in L 0,1 2 , and {β k (t)} is a sequence of mutually independent standard scalar Wiener process in the probability space (Ω,F ,P). Then T rQ = T rφφ * = In addition, we use C b (L 2 (D)) to denote C b (L 2 (D); R), the set of bounded continuous function mapping L 2 (D) to R. As in Da Prato and Zabczyk [7], for µ ∈ M(L 2 (D)) and ϕ ∈ C b (L 2 (D)), let P * t denote a measure semigroup mapping from M(L 2 (D)) to M(L 2 (D)), and P t be a transition probability semigroup, which are defined as A probability measure µ ∈ M(L 2 (D)) is called an invariant measure if P * t µ = µ for all t ≥ 0. Using the same arguments as Lv and Duan [19], and Brzezniak, Debbi and Goldys [2], it is easily to derive the well-posedness and the existence of invariant measure of SFBE (1) as follows.
3. Ergodicity. Since SFBE (1) is driven by the degenerate noise, we study asymptotically strong Feller property originated from Hairer and Mattingly [14], instead of the strong Feller property of the transition probability semigroup P t defined as (8). In this section, we also denote u(t, ω; u 0 ) = Φ t (ω, u 0 ) of SFBE (1).
For 0 ≤ r < t, let J r,t ξ be the solution of the linearized equation When r = 0, for simplicity, we write J t ξ = J 0,t ξ. It is not difficulty to show that for every ω, Given v ∈ L 2 loc (R + , R 1 ), The Malliavin derivative of Φ t (ω, u 0 ) with respect to ω in the direction v is given by By Duhamel principle, Lemma 3.1. Let s be in ( 1 2 , 1) and u 0 be in V . There then exist two positive constants C and η * > 0 such that for every t > 0 and every η ∈ (0, η * ], Proof. Using Itô's formula and noticing that (−∆) s u, u = u 2 V and uu x , u = 0, we have which implies that Since there exists a positive constant α > 0 such that u(r) 2 > α 2 Q * u(r) 2 , it yields that Further using the inequality [20, Lemma A.1] and taking the parameters β → ∞ and γ = 1 in the inequality, we drive which, similarly as the proof of Lemma A.1 in Hairer and Mattingly's [14], immediately implies Further note that if P(X ≥ C) ≤ 1 C 2 for any C > 0, then EX ≤ 2. Taking η * = α 2 and C = 2e αN 2 , the proof is completed.
Lemma 3.2. Let s be in ( 1 2 , 1) and u 0 be in V 1 . For any T > 0, there exists a positive constant C and η * > 0 such that every η ∈ (0, η * ], Proof. It follows from Itô's formula that Noting that u(t, x) = 0, as x ∈ D c , we have Meanwhile, from the Höder's inequality and the Young's inequality, we get Hence, we can deduce that which, similarly as the proof of Lemma 3.1, immediately implies the third result holds. Furthermore, if taking the expectation, we obtain which implies from the Gronwall's inequality that the first and the second results hold. The proof is completed.
Lemma 3.3. Let s be in ( 1 2 , 1) and u 0 be in V 1 . For any T > 0, there exists a positive constant C and η * > 0 such that every η ∈ (0, η * ] and t ∈ (0, T ] Proof. From the proof of Lemma 3.2, we can deduce that Then taking the expectation, we get If put N t := 2η where N t is the quadratic variation of N t . Noticing that Lemma 3.2, we know Ee H 1 dr ≤ C, satisfying the Novikov condition [22], which implies that e Mt is an exponential martingale. Then Ee Mt = Ee M0 = 1, which, therefore, from (15) and Lemma 3.2 that Lemma 3.3 holds. The proof is completed. Now, we begin to prove the asymptotical gradient estimate for the transition probability semigroup P t . Proposition 3.4 (Asymptotical gradient estimate). Let P t be the transition probability semigroup of SFBE (1). There exist some finite integer N * ∈ N and constants C, δ > 0 such that for any Frechét differentiable function ϕ, Proof. For any ξ = ξ l + ξ h ∈ H with ξ = 1. We further define Let ζ h (t) satisfy the equation Set ζ(t) := ζ l (t) + ζ h (t). It is clear that ζ(t) and (t) satisfy the same equation (11) with the same initial data (0) = ζ(0) = ξ, which implies that = ζ.
For simplicity, we also put G(t) : In the following, we firstly show that there exist constants δ > 0 and C > 0 such that For the low frequency ζ l (t), it obviously satisfies ζ l (t) ≤ 1 for 0 ≤ t ≤ 2 and ζ l (t) = 0 for t ≥ 2.
For the high frequency ζ h (t), we have from (17) that Next, we will estimate R 1 and R 2 . Firstly, we estimate R 1 .
For J 1 , by the Hölder's inequality and the Young's inequality, we get that For J 2 , applying Lemma 2.6 with s 1 = 1, s 2 = 0, s 3 = s ∈ (0, 1) and d = 1, and the Young's inequality, noticing that W s,2 ρ (D) ⊆ W s,2 (D), it infers that Therefore, it holds that We estimate R 2 in the following. Because of that using the Hölder's inequality, the Gagliardo-Nirenberg's inequality and the Young's inequality, thanking to W s,2 ρ (D) ⊆ W s,2 (D), we have Especially, take C 2 2 + C 4s· 4s−1 3 ≤ 2. Then by the Gronwall's inequality, we obtain Furthermore, it holds that Taking expectation and using Lemma 3.1 and Lemma 3.3, it infers that Therefore, combining with E ζ l (t) 2 ≤ C, then the above estimate (18) holds. Secondly, we show that Indeed, since v(t) = Q −1 G(t), we only need to show t 0 E G(r) 2 dr ≤ Ce C u0 2 . By definition of G(t), it deduces that Further, it follows from Lemma 2.7 that π l (uζ) x ≤ π l (uζ x ) + π l (ζu x ) ≤ C u ζ , which implies from Lemma 3.1, (18) and (21) that (20) holds. Finally, let's prove the main result (16) in Proposition 3.4. Using the chain rule and the integration by parts formula, we have Now, since that v [0,t] is adapted to the Wiener path, it follows from (20) that Consequently, it yields from (18) and (22) that (16) holds, which completes the proof.
In the following, we consider the irreducibility of the transition probability semigroup P t of SFBE (1). Proposition 3.5. Zero belongs to the support of any invariant measure of {P t } t≥0 .
Give arbitrary T > 0 and ε > 0, which are choosed in the later. We assume that It follows from (23) that By the Hölder's inequality and that W 1+s,2 ρ (D) ⊆ W s,2 ρ (D), using the Young's inequality, it implies that Applying integration by parts, we have Applying the Hölder's inequality, the Young's inequality and integration by parts, one has Furthermore, it implies that Hence, we get the following estimate from the above estimates, where γ = 2 − Cε − ε. Therefore, we obtain that for any ε , h > 0, there exist T > 0 and ε small enough such that sup and sup By the Gronwall's inequality, one has v(t) 2 ≤ e −γ(t−r) v(r) 2 + Cε.
Setting r = 0 and t = T we have v(T ) 2 ≤ C.
Then setting r = T and t = T + h, we get that v(T + h) 2 ≤ Ce −γh + Cε, which together with (27) yields that there exist a T large enough and a ε small enough such that v(T ) ≤ r 2 2 .
Similarly as Hairer and Mattingly's work [15], combining with Proposition 3.4 (asymptotical gradient estimate) and Proposition 3.5 (irreducibility), we only need to look for a continuous function V (u) to satisfy the Assumption 4 in the work [15]. In fact, if taking V (u) = exp(η ∇u 2 ), using Lemma 5.1 of [15], and combining the proof of Lemma 3.2 with the estimating of Eq.(9), we easily examine it. Consequently, the exponential ergodicity of the system SFBE (1) still holds.