STOCHASTIC BURGERS EQUATIONS WITH FRACTIONAL DERIVATIVE DRIVEN BY FRACTIONAL NOISE

. In this article, we study fractional stochastic Burgers equations perturbed by fractional noise. Existence and uniqueness of a mild solution is given by a ﬁxed point argument. Then, we explore H¨older regularity of the mild solution in C ([0 ,T ∗ ]; L p (Ω; ˙ H γ )) for some stopping time T ∗ .

The classical stochastic Burgers equation models a turbulent flow and is solved by the Hopf-Cole transformation [5,6,7].
In the past few years, stochastic Burgers equations perturbed by different random noises have been studied intensively. This equation plays an important role in nonlinear acoustics, cosmology, and statistical physics [1,2,3,21]. The author in [16] considers one dimensional stochastic Burgers equation driven by white noise term, and obtains existence of a weak solution by proving tightness for a sequence of polygonal approximations and solving a martingale problem for the weak limit.
In [9] the authors explore the existence and uniqueness of the global solution of a stochastic Burgers equation perturbed by white noise, and the existence of an invariant measure the corresponding transition semigroup. Stochastic Burgers equations with fractional Laplacian in spatial variable have been also explored. For instance, the researchers in [25] study a model involving the Lipschitz continuity of the inhomogeneous term, and a diffusion coefficient with space-time white noise in local subspace. In [4] the authors explore existence and uniqueness of invariant measures for the stochastic Burgers equation driven by fractional Laplacian and space-time white noise. They show that the transition measures of the solution converge to the invariant measure in the norm of total variation.
Many researchers have developed interests in the time-fractional diffusion equations [24,27,30] which are also applied for describing the memory effect of the wall friction through the boundary layer [13]. The authors in [8] studied the nonlinear stochastic equation of fractional derivative both in space and time variables with space-time white noise. They obtained the existence and uniqueness of solution with the moment bounds of solutions under Dalang's condition. In [31], it is proved that there is a unique mild solution of the stochastic Burgers equation with time-and space-fractional derivative driven by white noise, by a Picard iteration method. Different from the white noise in the model in [31], here we consider the fractional noise in time variable.
The fractional Brownian motion was first introduced with a Hilbert space framework by Kolmogorov in [17]. In recent years, fractional Brownian motion has been attracted attention because of their useful feature of preserving long term memory, and a large number of interesting results from scaling invariance to the description of their laws as random fields have been established by various authors. The study of these Gaussian processes has its historical motivation from their applications in hydrology and telecommunication, and has been applied to the mathematical finance, biotechnology and biophysics, see for example [11,18,23] and their references. In [14], the researchers explore that the existence, uniqueness, and moment estimate for the solution of the stochastic Burgers equation driven by multi-parameter fractional noise. The authors in [26] show the local and global existence and uniqueness results for the stochastic Burgers equation driven by fractional Brownian motion with H > 1 4 . In our work, we consider a U-valued Q-cylindrical fractional Brownian motion with Hurst parameter H ∈ (1/2, 1).
The above research work motivates us to obtain the existence and uniqueness of the mild solution to the problem (1.1) with boundary and initial conditions, and explore Hölder regularity of the mild solution.
Definition 1.1. The domain of the fractional Laplace operator (−∆) α/2 in (1.2) is defined asḢ with the inner product ·, · in L 2 (D). Thus, we define the norm as (1.4) In this article we use the following notation.
. Thus, problem (1.1) can be rewritten as Now, we introduce the Bochner spaces L p (Ω; G) = L p ((Ω, F , P); G) as with the norm f L p (Ω;G) = (E f p G ) 1/p , where G is a Banach space. Next, we define mild solutions of problem (1.5), which is inspired by the definition of mild solution to the fractional stochastic Burgers equations driven by multiplicative white noise [31]. Definition 1.2. Let {u(t), t ∈ [0, T ]} be a random field that is continuous with respect to t. A function u ∈ C([0, T ]; L p (Ω;Ḣ γ )) is a mild solution of (1.5) if where L α β (t) and L α β,β (t) are the generalized Mittag-Leffler operators defined by (2.5) and (2.6).
A derivation of the mild solution is shown in the Appendix, which applies the Laplace transform method and the properties of the semigroup generated from the fractional Laplace operator. In the following, we give some assumptions about the operator B and the initial condition u 0 .
for all u, v ∈ L 2 (D), where M is a positive constant.
Assumption 1.4. Let the initial value u 0 : Ω →Ḣ γ be F 0 -measurable random variable, satisfying We set a subspace of C([0, T ]; L p (Ω;Ḣ γ )) for a stopping time T as follows where γ > 0, 0 < T T . The main results in this article reads as follows.

2β
. Let u be a solution of (1.1) in S T * , with T * satisfying the conditions in Theorem 1.5. Then for any 0 t 1 < t 2 T * , the solution u(t) is Hölder continuous with respect to the norm · L p (Ω;Ḣ γ ) and satisfies Now we highlight the contribution of this article in the field of the fractional stochastic Burgers equations. Firstly, our model with time-and space-fractional stochastic Burgers equation driven by fractional noise is new, compared to the problems studied in [4,9,14,31]. Secondly, the U-valued Q-cylindrical fractional Brownian motion makes some difficulties in the analysis, we apply the embedding theorem to solve these difficulties. Finally, we compose the fractional Laplacian and the generalized Mittag-Leffler operators to estimate the norm of the mild solution of problem (1.1).
This article is organized as follows. In Section 2, we present some notation and introduce fractional Brownian motion, the generalized Mittag-Leffler operators. Then we give properties of fractional Laplacian and the generalized Mittag-Leffler operators. In Section 3, we prove Theorem 1.5 to obtain the existence and uniqueness of mild solution by the Banach Fixed Point Theorem for some stopping time. In Section 4, we prove Theorem 1.6, to obtain the Hölder continuity of the mild solution finally.

Fractional Brownian motion.
We provide an overview and systematization of stochastic calculus with respect to fractional Brownian motion. First, we introduce the one-dimensional fractional Brownian motion briefly; see [22] for details. A one-dimensional fractional Brownian motion with Hurst parameter H ∈ (0, 1) is a centered Gaussian process B H := {B H (t), t ≥ 0} with the covariance function Note that B 1/2 (t) is standard Brownian motion. We denote by E the set of step functions on [0, T ]. Let H be the Hilbert space defined as the closure of E with respect to the scalar product The mapping 1 [0,t] → B H (t) can be extended to an isometry between H and the Gaussian space associated with B H . When H > 1 2 , it has been proved the covariance of fractional Brownian motion can be written as Consider the square integrable kernel where t > s > 0, B(·, ·) is the beta function. We deduce that this kernel satisfies From the definition of K H , we obtain that We consider the linear operator K * H from E to L 2 ([0, T ]) defined by H is an isometry between E and L 2 ([0, T ]) that can be extended to the Hilbert space H. In fact, for any s, t ∈ [0, T ] we have Since the operator K * H provides an isometry between the Hilbert space H and L 2 ([0, T ]), it follows that for any t ∈ [0, T ] there exists a Brownian motion Moreover, for any ϕ ∈ H, we have Next we introduce the fractional Brownian motion with values in a Hilbert space and give the definition of the corresponding stochastic integral. Let U , V be separable Hilbert spaces, and L(U, V ) denote the space of all bounded linear operators from U to V . Let Q ∈ L(U, U ) be a nonnegative self-adjoint operator, and let {σ n } n∈N be a bounded sequence of nonnegative real numbers such that Qς n = σ n ς n with ∞ n=1 σ n < ∞, where {ς n } n∈ N is a complete orthonormal basis in U . We denote by L Q (U, V ) the space of all ϕ ∈ L(U, V ) such that ϕQ 1/2 is a Hilbert-Schmidt operator with the norm Then ϕ is called a Q-Hilbert-Schmidt operator from U to V . Let {B H n (t)} n∈N be a sequence of two-sided one-dimensional standard fractional Brownian motions mutually independent on (Ω, F , P). When one considers the series which not necessarily converges in the space U . Then we consider the U -valued stochastic process Since Q is a nonnegative self-adjoint operator, the above series converges in the space U , that is, it holds that W H (t) ∈ L 2 (Ω, U ). Then, we say that W H (t) is a well defined U -valued Q-cylindrical fractional Brownian motion with covariance operator Q such that The following lemma estimates the stochastic integrals, see [29] for details.
where the constant C(H, p) is positive.

Fractional Laplace Operator.
For the operator A α introduced in the previous section, we have the following property; see [28].
For any α > 0, the operator −A α generates an analytic semigroup S α (t) = e −tAα , t 0 on L 2 (D). And for each γ 0, there exists a constant C(α, γ) such that Here L(L 2 ) denotes the Banach space of linear bounded operators from L 2 (D) to itself.
In this article, the constant C is different from line to line.

2.3.
Mittag-Leffler Operator. In this subsection, we introduce the one-sided stable probability density function. For each β ∈ (0, 1), θ ∈ (0, +∞), there exists and the Mainardi's Wright-type function (see [19,20]) given by Thus, we can obtain the following properties The above Mainardi function M β (θ) acts as a bridge between the following generalized Mittag-Leffler operators and the fractional differential equation (1.5). The generalized Mittag-Leffler operators are defined as Now we give some properties of these two operators; see [31].
Proof. We know from the properties of the semigroup S α (t) and A α that Since 0 < t 0 < t ≤ T , we can deduce that for each f ∈ L 2 (D),
Corollary 2.7. If we assume η = 0 in Lemma 2.6, then for each f ∈ L 2 (D), t ∈ (t 0 , T ], we have Following as similar method as in Lemma 2.6, we have that

Mild solution
In this section, we prove, Theorem 1.5, the existence and uniqueness of a mild solution of (1.1), by the Banach Fixed Point Theorem for some stopping time T * in the space Proof of Theorem 1.5. We define a map F : S T → C([0, T ]; L p (Ω;Ḣ γ )) for u ∈ S T as follows Firstly, we show that the map F is well defined. Indeed, for any u ∈ S T , from f Ḣγ = A γ f in (1.4) and the definition of the operators L α β and L α β,β in (2.5) and (2.6), we have that From the properties of A α in Lemma 2.3, and Assumption 1.4, we deduce that From the properties of L α β,β (t) in Lemma 2.5, we have Since u ∈ S T , by Assumption 1.3, B(u) M u 2 , and the Hölder inequality, (3.4) implies that (3.5) It follows that with 0 < γ < α(pβ−1) pβ , where λ 1 and Λ are the minimum and maximum of the eigenvalues of the operator −∆ relatively in the notation in (1).
Secondly, we want to find T 0 ∈ (0, T ] such that F : S T0 → S T0 . By the same arguments as the above analysis (3.3)-(3.7), we obtain that where the constant C is positive and depends on α, β, γ, H, R 0 , T, M, K, λ 1 , Λ.
where γ such that 0 < γ < α(2β−1) pβ . Finally, we show F is a contraction mapping on S T * with suitable selected T * such that 0 < T * < T 0 . For any u, h ∈ S T0 , taking similar method as in the estimate (3.4)-(3.6), and by Assumption 1.3 and Hölder inequality, we have that where Λ, λ 1 are the maximum and minimum of the eigenvalues of (−∆) relatively. Then, it further implies that with 0 < γ < α(2β−1) We take T * ∈ (0, T 0 ) such that By the Banach Fixed Point Theorem, there exist a unique point u ∈ S T * , which is a unique mild solution to the problem (1.5). Then by the equivalency of the problem (1.5) and (1.1), the Theorem 1.5 is proved.

Hölder continuity
In this section, we prove Theorem 1.6, and obtain the Hölder continuity of the mild solution in (1.1).
Proof of Theorem 1.6. For any 0 t 1 < t 2 T * , since u is a mild solution of (1.5), we have  Firstly, we consider the term I 1 . From Lemma 2.6 and Assumption 1.4, we deduce that Secondly, for the term I 2 , we divide it into three parts as follows  For I 21 , by Assumption 1.3 and Lemma 2.6, we have Then, by using the Hölder inequality, we obtain where the last inequality in (4.5) holds because u ∈ S T * and 1/2 < β < 1, p > 2. Thus, we have Next we estimate I 22 and I 23 similarly as for (4.4) and (4.5). By applying Lemma 2.5, we can deduce that with 0 < γ < α(pβ−1) pβ and p > 2, and From (4.3)-(4.7), we obtain where C(α, β, γ, λ 1 , Λ, M, K, T * , p) is a positive constant. Finally, we estimate the term I 3 in (4.1). Here as in (4.3), the term I 3 is divided into three parts, .

Appendix
Here, we give the derivation of the mild solution of the abstract problem (1.6); for more details see [31].    Based on estimates (5.2)-(5.4) and using the inverse Laplace transform, we obtain that the mild solution satisfies