A spatial version of the It\^{o}-Stratonovich correction

TL;DR

This paper investigates how spatial discretization of stochastic Burgers equations driven by space-time white noise introduces a correction term analogous to the Itô-Stratonovich correction, due to local quadratic cross-variation effects.

Contribution

It reveals that improper spatial discretization leads to a correction term in the limit, linking spatial quadratic variation to stochastic PDE discretization effects.

Findings

Discretization errors induce a correction term proportional to local quadratic cross-variation.

The correction term is analogous to the classical Itô-Stratonovich correction in stochastic calculus.

The limit behavior depends on the discretization scheme used.

Abstract

We consider a class of stochastic PDEs of Burgers type in spatial dimension 1, driven by space-time white noise. Even though it is well known that these equations are well posed, it turns out that if one performs a spatial discretization of the nonlinearity in the "wrong" way, then the sequence of approximate equations does converge to a limit, but this limit exhibits an additional correction term. This correction term is proportional to the local quadratic cross-variation (in space) of the gradient of the conserved quantity with the solution itself. This can be understood as a consequence of the fact that for any fixed time, the law of the solution is locally equivalent to Wiener measure, where space plays the role of time. In this sense, the correction term is similar to the usual It\^{o}-Stratonovich correction term that arises when one considers different temporal discretizations of stochastic ODEs.