Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion

In this paper, we consider the numerical approximation for a class of fractional stochastic partial differential equations driven by infinite dimensional fractional Brownian motion with hurst index \begin{document}$ H∈ (\frac{1}{2}, 1) $\end{document} . By using spectral Galerkin method, we analyze the spatial discretization, and we give the temporal discretization by using the piecewise constant, discontinuous Galerkin method and a Laplace transform convolution quadrature. Under some suitable assumptions, we prove the sharp regularity properties and the optimal strong convergence error estimates for both semi-discrete and fully discrete schemes.


1.
Introduction. In recent years, there has been considerable interest in studying fractional stochastic partial differential equations (SPDEs, in short) due to their applications in various scientific and technological areas including physics, biology, telecommunications, turbulence, and engineering (see, for example, Mehaute [21]). For example, the classical heat equation ∂ t u(t, x) = ∆u(t, x), is used for modeling heat diffusion in homogeneous media, while the fractional heat equation ∂ β t u(t, x) = ∆u(t, x), describes heat propagation in inhomogeneous media. Moreover, stochastic partial differential equations driving by infinite dimensional fractional Brownian motion also is a recent research direction in probability theory and its applications, see, for examples [4,18,8,7,30].
On the other hand, numerical analysis of stochastic partial differential equations is currently an active area of research due to the need of practical applications, and there are a large number of literatures on numerical methods for spdes. So far, most of the work done on the stochastic evolution equations has dealt with [ΨẆ H (t)], α ∈ (0, 1), t ∈ [0, T ], where Ψ : U → U is deterministic mapping, I 1−α t is the fractional integral operator defined by t 0 (t − r) −α f (r)dr, α ∈ (0, 1), and the fractional differential operator D α t is understood in the Caputo sense (see, Caputo [2]) dr (t − r) α , α ∈ (0, 1).
In the past years, many researchers have investigated the numerical approximation of deterministic fractional partial differential equations by different methods. For example, an implicit finite difference method has been used in [5,26,36] while a standard Galerkin finite element method has been employed in [19,20] . Recently, Li and Yang [17] studied the numerical approximation for fractional SPDEs driven by Q-Wiener process, where they used the standard continuous finite element method and the piecewise constant, discontinuous Galerkin method to obtain a fully discrete implicit scheme and get the the corresponding strong convergence error estimates, respectively. However, to the best knowledge of the authors, there is little work on the numerical approximation for SPDEs driven by fractional Brownian motion. In this paper, our aim is to discuss the numerical approximation for fractional SPDEs driven by fractional Brownian motion and obtain optimal strong convergence error estimates for both semi-discrete and fully discrete schemes under some suitable assumptions. The rest of the paper is organized as follows. In Section 2, we recall some properties of the stochastic integration with respect to W H and give the solution representation of (1) by means of the Mittag-Leffler function. In Section 3, we obtain the sharp regularity of the equation (1). Section 4 is devoted to study the optimum error estimates both in space and time with smooth initial data.
2. Preliminaries. In this section, we briefly recall the definition of the stochastic integration with respect toẆ H and the representation of the solution to equation (1) by using the Mittag-Leffler function. For convenience, we let C stand for a positive constant depending only on the subscripts, while its value may be different in different appearance.
Throughout this paper we assume that U is a real separable Hilbert space with the inner ·, · and the norm · . Moreover, let L (U ) be the space of bounded linear operators from U to U endowed with the usual operator norm · L (U ) and let L 2 (U ) ⊂ L (U ) be the space of all Hilbert-Schmidt operator equipped with the inner product and norm where {e n } n∈N is a complete orthonormal basis of U . Moreover, ·, · L2(U ) and · L2(U ) are independent of the choice {e n } n∈N (see, DaPrato and Zabczyk [6]).

Cylindrical fractional Brownian motion. Recall that a real valued Gauss
where {w H n (t)} t∈[0,T ] is a sequence of mutually independent real-valued standard fBms each with the same Hurst index 0 < H < 1.
We now define the stochastic integration of an integrand Φ : [0, T ] → L 2 (U ) with respect to cylindrical fBm W H , denoted by There are different ways to define I(Ψ(s)), here we adopt the method in Duncan et al [8] and recall an inequality of the integrands. Lemma 2.1 (Duncan et al [8]). Let f ∈ L p (0, T ; R) for p > 1 H be a deterministic function. Then there exists a constant C T > 0 such that where φ(u, v) = α H |u − v| 2H−2 with α H = H(2H − 1).

LITAN YAN AND XIUWEI YIN
Let E be the family of U -valued step functions, defined by It is easy to see that . Since E is dense in L p (0, T ; U ), the stochastic integration defined above can be uniquely extended to L p (0, T ; U ). To define the integral I(Ψ(s)), we also need the next assumptions: Under these assumptions, we define the stochastic integral I(Ψ(s)) by where the summation is in L 2 (Ω) sence. The series in (2) is a zero mean, U -valued Gaussian random variable because of 2.2. The representation of a solution to (1). In order to give the representation, we need some assumptions. Under this assumption, there exists an increasing sequence of real numbers {λ n } n∈N , and an orthonormal basis {e n } n∈N of U such that Ae n = λ n e n and 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ n ≤ · · · → ∞, n → ∞.
Then, we can define the fractional powers of A, i.e. A γ , γ ∈ R as follows: LetU γ = D(A γ/2 ) and its norm is denoted by Consider the operator E(t) by , z ∈ C, β ∈ R is the Mittag-Leffer function. By using time fractional Duhamel's principle (see, for example, Umarov [32]), we can define the solution X(t) of (1) as follows The function E α,β (z) and the operator E(t) admit the following properties.

Lemma 2.3 ([27]
). For any α, λ > 0, and m ∈ N, we have 3. Sharp regularity results. In this section, we prove the regularity properties of equation (1) in both time and space. Keeping the notations in Section 2. The following assumption will be used.

LITAN YAN AND XIUWEI YIN
Example 3.1. Consider the following fractional stochastic heat equation: Here U = L 2 (0, 1), Ψ = I U and −A is the Laplacian operator with Dirichlet boundary conditions. It is well known that Thus, Assumption 3.1 is satisfied.
We firstly estimate the term I 1 , according to lemma 2.2, x, e j 2 .
We now turn to estimate I 2 , using a change of variable and combine with lemma 2.2, we have where we have used the fact that for H < α < 1 (see Nualart [24]), Combining the above two estimates, we complete the proof.
For the term Λ 1 , by the proof of Lemma 4.3 in Li and Yang [17] we have which implies that x, e j 2 .
On the other hand, Lemma 2.2 deduces to x, e j 2 .
The proof is completed.
Then, under assumptions 2.1 and 3.1, the stochastic evolution is well-defined in L 2 (Ω, U ) and admits the following temporal and space regularity: (ii) for any ρ ∈ [0, 2H+αβ−1 α ], the following temporal regularity holds: Proof. The well-posedness of Ξ(t) is clear for every t ∈ [0, T ], and we only need to check the temporal and space regularity. We first prove the statement (i). By lemma 3.1 it follows that .
We now prove the statement (ii). Observe that To estimate I 3 , for 0 < s < t, we have For the term Θ j (j ≥ 1), by integration term-by-term we have with 0 < s < t, i.e., tE α,2 (−λt α ) ≥ sE α,2 (−λs α ), 0 < s ≤ t. It follows that for any 0 < t < t. Then we have This gives the desired estimate . Finally, we can easily get that for all 0 < s < t. Thus, we have completed the proof of the statement (ii).
As an immediate conclusion of Proposition 1 and Lemma 3.2, we can establish the following result. We omit the proof of the existence and uniqueness of the mild solution as it is standard. For more details, we refer the readers to [6,16].
and for any ρ ∈ [0, (2H + αβ − 1)/α], It is important to note that the equation (1) reduces to the abstract evolution equation driven by Cylindrical fractional Brownian motion, as α → 1, which was investigated in Wang et al [34], and Theorem 3.3 coincides with Theorem 3.5 in Wang et al [34]. In order to motivate why we discuss the optimal spatial regularity, we conclude this section with the following example, which is a slight modification in the example from Wang et al [34]. More precisely, denote U = L 2 (0, 1) and let −A be the Laplacian with Dirichlet boundary conditions. Define the stochastic evolution

OPTIMUM ERROR ESTIMATES FOR FSPDE'S WITH FBM 625
where Q : U → U is defined by = ∞ for any δ > β. On the other hand, for any γ > 0 and t > 0, 4. Optimal error estimate of a full-discretization. In this section, we obtain the optimal convergence rates for strong approximate of the underlying problem, which coincides with the regularity of the solution.
4.1. Spatial semi-discretization. We spatially discretize (1) with a spectral Galerkin method (for more details, see Thomée [29]). Let N ∈ N and define a finite dimensional subspace of U by U N := span{e 1 , . . . , e N }. Define the projection P N : U ς → U N by x, e i e i , ∀ x ∈ U ς , ς ∈ R.
It is easy to see that where A N : U → U N , A N := AP N . Then, the mild solution of (6) is The following theorem gives the error estimate for the spectral Galerkin discretization (6).

LITAN YAN AND XIUWEI YIN
for all N ∈ N.
Proof. Observe that It is easy to check that Now, we turn to estimate Λ 4 . Define It follows that for any x ∈ U αβ−1 α , which implies that Thus, the lemma follows from (8), (9) and (10).

4.2.
Fully discrete error estimate. This subsection devotes to a fully discrete approximate for equation (1). We firstly consider time discretization corresponding to equation (1) and analysis the corresponding error estimate. By operating D 1−α t on both sides of (1), we get where D t denotes the usual derivative with respect to t. For a fixed time step size ∆t > 0, we put t n = n∆t and time discretization (11) by using the DG method (see [19,20]), define a piecewise constant approximation V (t) ≈ X(t) by denotes the one-side limit from below at the nth time level. Thus, V (t) = V n , t n−1 < t ≤ t n . An elementary calculus may prove that tn tn−1 Following Kovács and Printems [14] (see also [17]), we can give a formulation for the discrete mild solution of (12). Consider the deterministic algorithm k=0 β k z k , then by taking z-transform we have That is, We can rewrite B(z)x 0 as (see (4.4) in [17]) Taking z-transform for (12), then Combining this equality with the expression of B(z)x 0 , we can easily get that Taking the inverse z-transform, we further get . . , n. Then, for any x ∈ U , we have Proof. Denote ζ(u, i) = |E α,1 (−λ m (t n −u) α )−E α,1 (−λ m (t n −t i ) α )| for i = 0, 1, . . . , n, and It follows that x, e m On the other hand, by the fact where t i ≤ u ≤ t i+1 and 0 < δ ≤ 1. Combining this with a change of variables, we see that For the term Λ 6 , in a similar argument as Lemma 3.1, we get Thus, the lemma follows from (14).
Proof. We have It follows from Lemma 4.1 in [17] that We now turn to estimate Λ 10 . Let [t] := t i , t ∈ [t i , t i+1 ),Ẽ(t) = B j and t ∈ [t j−1 , t j ). Then, we have By Lemma 4.2, it follows that .
Similarly, by lemma 4.3 we also have .
Thus, we have gotten the desired estimate , and the Theorem follows from (17) and (18). Now, we consider a fully discrete approximation for equation (1). Let  On the one hand, the estimation is a direct consequence of Theorem 4.1. On the other hand, it follows from Theorem 4.4 and the contractive property of P N that Therefore the proof is completed.