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

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.