Dynkin's isomorphism theorem and the stochastic heat equation
Résumé
Consider the stochastic heat equation partial derivative(t)u = Lu + W, where 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 L. In the case that L is the generator of a Levy process on R (d) , our result gives a probabilistic explanation of the recent findings of Foondun et al. (Trans Am Math Soc, 2007).