Approximating rough stochastic PDEs

TL;DR

This paper investigates the convergence of various spatial approximations to vector-valued Burgers-type stochastic PDEs driven by white noise, revealing that the limit can depend on the approximation scheme, similar to Itô-Stratonovich corrections.

Contribution

It introduces a detailed analysis of approximation schemes for rough stochastic PDEs, showing convergence and highlighting the dependence of limits on the approximation method.

Findings

Approximations to rough stochastic PDEs converge under certain conditions.

The limit of approximations can vary depending on the scheme used.

Convergence rates are established for these approximation methods.

Abstract

We study approximations to a class of vector-valued equations of Burgers type driven by a multiplicative space-time white noise. A solution theory for this class of equations has been developed recently in [Hairer, Weber, Probab. Theory Related Fields, to appear]. The key idea was to use the theory of controlled rough paths to give definitions of weak / mild solutions and to set up a Picard iteration argument. In this article the limiting behaviour of a rather large class of (spatial) approximations to these equations is studied. These approximations are shown to converge and convergence rates are given, but the limit may depend on the particular choice of approximation. This effect is a spatial analogue to the It\^o-Stratonovich correction in the theory of stochastic ordinary differential equations, where it is well known that different approximation schemes may converge to different solutions.