Potential Analysis

, Volume 34, Issue 3, pp 243–260 | Cite as

Dynkin’s Isomorphism Theorem and the Stochastic Heat Equation

  • Nathalie Eisenbaum
  • Mohammud Foondun
  • Davar Khoshnevisan


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).


Stochastic heat equation Local times Dynkin’s isomorphism theorem 

Mathematics Subject Classifications (2010)

60J55 60H15 


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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

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