Uniqueness in Law for the Allen-Cahn SPDE via Change of Measure

TL;DR

This paper establishes a novel method using change of measure to transfer uniqueness in law among various SPDEs, notably proving uniqueness for the Allen-Cahn SPDE with specific diffusion functions under weaker conditions than traditional approaches.

Contribution

It introduces a new approach to prove uniqueness in law for SPDEs using a weaker $L^2$ condition, extending results to the Allen-Cahn SPDE and other equations.

Findings

Proves transfer of uniqueness in law under weaker conditions than Novikov.

Establishes uniqueness in law for Allen-Cahn SPDE with specific diffusion functions.

Applicable to SDEs and hyperbolic SPDEs as well.

Abstract

We start by first using change of measure to prove the transfer of uniqueness in law among pairs of parabolic SPDEs differing only by a drift function, under an almost sure $L^2$ condition on the drift/diffusion ratio. This is a considerably weaker condition than the usual Novikov one, and it allows us to prove uniqueness in law for the Allen-Cahn SPDE driven by space-time white noise with diffusion function $a(t,x,u)=Cu^\gamma$, $1/2\le\gamma\le1$ and $C\ne0$. The same transfer result is also valid for ordinary SDEs and hyperbolic SPDEs.