The Stratonovich heat equation : a continuity result and weak approximations

TL;DR

This paper establishes the existence, uniqueness, and continuity of solutions to a nonlinear Stratonovich heat equation with multiplicative noise, and demonstrates weak convergence of noise approximations using rough path techniques.

Contribution

It introduces a novel continuity result for the solution with respect to noise approximations in the context of a nonlinear Stratonovich heat equation.

Findings

Proved unique mild solution existence.

Established almost sure continuity with respect to noise.

Demonstrated weak convergence via Donsker and Kac-Stroock approximations.

Abstract

We consider a Stratonovich heat equation in $(0,1)$ with a nonlinear multiplicative noise driven by a trace-class Wiener process. First, the equation is shown to have a unique mild solution. Secondly, convolutional rough paths techniques are used to provide an almost sure continuity result for the solution with respect to the solution of the 'smooth' equation obtained by replacing the noise with an absolutely continuous process. This continuity result is then exploited to prove weak convergence results based on Donsker and Kac-Stroock type approximations of the noise.