SPDEs law equivalence and the compact support property: applications to the Allen-Cahn SPDE

TL;DR

This paper establishes law equivalence between certain SPDEs, including the Allen-Cahn and heat SPDEs, and explores conditions for the compact support property, advancing understanding of their probabilistic behavior.

Contribution

It introduces a law transfer method for SPDEs, proving equivalence of laws for equations differing by drift and applying it to the Allen-Cahn SPDE.

Findings

Law equivalence between Allen-Cahn and heat SPDEs.

Criteria for the compact support property in Allen-Cahn SPDEs.

Extension of law transfer results to a broad class of SPDEs.

Abstract

Using our uniqueness in law transfer result for SPDEs, described in a recent note, we prove the equivalence of laws of SPDEs differing by a drift, under vastly applicable conditions. This gives us the equivalence in the compact support property among a large class of SPDEs. As an important application, we prove the equivalence in law of the Allen-Cahn and the associated heat SPDEs; and we give a criterion for the compact support property to hold for the Allen-Cahn SPDE with diffusion function $a(t,x,u)=Cu^\gamma$, with $C\ne0$ and $ 1/2\le\gamma<1$.