Potential Analysis

, Volume 34, Issue 3, pp 243–260

Dynkin’s Isomorphism Theorem and the Stochastic Heat Equation

  • Nathalie Eisenbaum
  • Mohammud Foondun
  • Davar Khoshnevisan
Article

DOI: 10.1007/s11118-010-9193-x

Cite this article as:
Eisenbaum, N., Foondun, M. & Khoshnevisan, D. Potential Anal (2011) 34: 243. doi:10.1007/s11118-010-9193-x
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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 Rd, our result gives a probabilistic explanation of the recent findings of Foondun et al. (Trans Am Math Soc, 2007).

Keywords

Stochastic heat equation Local times Dynkin’s isomorphism theorem 

Mathematics Subject Classifications (2010)

60J55 60H15 

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Nathalie Eisenbaum
    • 1
  • Mohammud Foondun
    • 2
  • Davar Khoshnevisan
    • 3
  1. 1.Laboratoire de Probabilités et Modèles AléatoiresCNRS, Université Paris VIParis Cedex 05France
  2. 2.School of MathematicsLoughborough UniversityLeicestershireUK
  3. 3.Department of MathematicsUniversity of UtahSalt Lake CityUSA