Localization errors in solving stochastic partial differential equations in the whole space

TL;DR

This paper demonstrates that localization of SPDEs on the whole space to finite domains results in exponentially small errors, enabling practical numerical schemes combining localization and discretization.

Contribution

It introduces a method to localize SPDEs on the whole space with exponentially small errors, facilitating implementable numerical schemes.

Findings

Localization error is exponentially small.

Numerical scheme combining localization and discretization is proposed.

Method enables practical computation for SPDEs on unbounded domains.

Abstract

Cauchy problems with SPDEs on the whole space are localized to Cauchy problems on a ball of radius $R$. This localization reduces various kinds of spatial approximation schemes to finite dimensional problems. The error is shown to be exponentially small. As an application, a numerical scheme is presented which combines the localization and the space and time discretisation, and thus is fully implementable.