Dynkin's isomorphism theorem and the stochastic heat equation

TL;DR

This paper explores the connection between the stochastic heat equation and Gaussian processes via Dynkin's isomorphism theorem, providing new insights into the structure of solutions related to Markov processes and Levy generators.

Contribution

It establishes a local absolute continuity between the solution of the stochastic heat equation and a Gaussian process linked through Dynkin's isomorphism, extending understanding to Levy processes.

Findings

Solution is locally mutually absolutely continuous with a Gaussian process

Provides probabilistic explanation for recent Levy process results

Connects stochastic heat equation solutions with local times of Markov processes

Abstract

Consider the stochastic heat equation $\partial_t u = \sL u + \dot{W}$, where $\sL$ is the generator of a [Borel right] Markov process in duality. We show that the solution is locally mutually absolutely continuous with respect to a smooth perturbation of the Gaussian process that is associated, via Dynkin's isomorphism theorem, to the local times of the replica-symmetric process that corresponds to $\sL$.In the case that $\sL$ is the generator of a L\'evy process on $\R^d$, our result gives a probabilistic explanation of the recent findings of Foondun et al.