, Volume 34, Issue 3, pp 243-260
Date: 03 Aug 2010

Dynkin’s Isomorphism Theorem and the Stochastic Heat Equation

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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).

Research supported in part by NSF grant DMS-0704024.