Approximating Stochastic Evolution Equations with Additive White and Rough Noises

TL;DR

This paper develops and analyzes Galerkin approximation methods for stochastic evolution equations driven by complex Gaussian noises, achieving optimal error estimates that improve upon existing numerical methods.

Contribution

It introduces a regularization approach using Wong-Zakai approximation and provides optimal error bounds for Galerkin discretizations of such equations.

Findings

Optimal convergence order for noise regularization.

Error estimates that eliminate infinitesimal factors.

Enhanced numerical accuracy for stochastic evolution equations.

Abstract

In this paper, we analyze Galerkin approximations for stochastic evolution equations driven by an additive Gaussian noise which is temporally white and spatially fractional with Hurst index less than or equal to $1/2$. First we regularize the noise by the Wong-Zakai approximation and obtain its optimal order of convergence. Then we apply the Galerkin method to discretize the stochastic evolution equations with regularized noises. Optimal error estimates are obtained for the Galerkin approximations. In particular, our error estimates remove an infinitesimal factor which appears in the error estimates of various numerical methods for stochastic evolution equations in existing literatures.