Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise

TL;DR

This paper analyzes the approximation quality of space discretizations for stochastic integral equations driven by Gaussian noise, focusing on fractional order PDEs and classical SPDEs, with error estimates linked to deterministic error rates.

Contribution

It provides new error estimates for space approximation of stochastic fractional PDEs using spectral Galerkin and finite element methods, improving existing results for the stochastic heat equation.

Findings

Error estimates in pathwise Hölder norms derived from deterministic error rates

Application to stochastic fractional PDEs with spectral Galerkin and finite element methods

Improvement of existing results on the stochastic heat equation

Abstract

We investigate the quality of space approximation of a class of stochastic integral equations of convolution type with Gaussian noise. Such equations arise, for example, when considering mild solutions of stochastic fractional order partial differential equations but also when considering mild solutions of classical stochastic partial differential equations. The key requirement for the equations is a smoothing property of the deterministic evolution operator which is typical in parabolic type problems. We show that if one has access to nonsmooth data estimates for the deterministic error operator together with its derivative of a space discretization procedure, then one obtains error estimates in pathwise H\"older norms with rates that can be read off the deterministic error rates. We illustrate the main result by considering a class of stochastic fractional order partial differential equations and space approximations performed by spectral Galerkin methods and finite elements. We also improve an existing result on the stochastic heat equation.