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).
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Research supported in part by NSF grant DMS-0704024.
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Eisenbaum, N., Foondun, M. & Khoshnevisan, D. Dynkin’s Isomorphism Theorem and the Stochastic Heat Equation. Potential Anal 34, 243–260 (2011). https://doi.org/10.1007/s11118-010-9193-x
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DOI: https://doi.org/10.1007/s11118-010-9193-x