Dynkin's isomorphism theorem and the stochastic heat equation

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 Lévy process on ${\mathbb{R}}^d$, our result gives a probabilistic explanation of the recent findings of Foondun et al.