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

TL;DR

This paper establishes sharp regularity results for semi-linear SPDEs driven by fractional Brownian motion with Hurst > 1/2 and determines optimal convergence rates for their spectral Galerkin and Euler numerical schemes.

Contribution

It provides the first sharp regularity results and optimal convergence rates for numerical approximations of SPDEs driven by fractional Brownian motion with Hurst > 1/2.

Findings

Sharp spatial and temporal regularity results obtained.

Optimal mean-square convergence rates identified.

Numerical examples confirm theoretical predictions.

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.