Sharp mean-square regularity results for SPDEs with fractional noise and optimal convergence rates for the numerical approximations

Abstract

This article offers sharp spatial and temporal mean-square regularity results for a class of semi-linear parabolic stochastic partial differential equations (SPDEs) driven by infinite dimensional fractional Brownian motion with the Hurst parameter greater than one-half. In addition, mean-square numerical approximation of such problem are investigated, performed by the spectral Galerkin method in space and the linear implicit Euler method in time. The obtained sharp regularity properties of the problems enable us to identify optimal mean-square convergence rates of the full discrete scheme. These theoretical findings are accompanied by several numerical examples.

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

“…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.…”

“…Some surveys and literatures can be found in Jin et al [12], McLean and Mustapha [19,20], Li and Yang [17]. However, in contrast to the extensive studies on fractional Brownian motion, there has been little systematic investigation on fractional stochastic partial differential equations driven by fractional Brownian motions (see, Kamrani and Jamshidi [13] and Wang et al [34]).…”

Optimal strong convergence rates of some Euler-type timestepping schemes for the finite element discretization SPDEs driven by additive fractional Brownian motion and Poisson random measure

“…In contrast to standard Brownian (H = 1/2) where there are numerous literature on numerical algorithms for SPDEs, few works have been done for numerical methods for fBm for SPDEs of type (5). Indeed, standard explicit and linear implicit schemes have been investigated in the literature for SPDEs of type (5) (see [13,14,38]). The works in [13,38] deal with self-adjoint operator and use the spectral Galerkin method for the spatial discretization.…”

“…Indeed, standard explicit and linear implicit schemes have been investigated in the literature for SPDEs of type (5) (see [13,14,38]). The works in [13,38] deal with self-adjoint operator and use the spectral Galerkin method for the spatial discretization. This is very restrictive as many concrete applications use non self-adjoint operators.…”

Analytical Solutions for Multi-Time Scale Fractional Stochastic Differential Equations Driven by Fractional Brownian Motion and Their Applications

“…X.J. Wang et al [32] considered the following semilinear parabolic SPDEs in V, driven by an infinite dimensional fractional Brownian motion,…”