Dynkin’s Isomorphism Theorem and the Stochastic Heat Equation

Abstract

Consider the stochastic heat equation $\partial_t u = \mathcal{L} u + \dot{W}$ , where $\mathcal{L}$ 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 $\mathcal{L}$ . In the case that $\mathcal{L}$ is the generator of a Lévy process on R d , our result gives a probabilistic explanation of the recent findings of Foondun et al. (Trans Am Math Soc, 2007).