Probabilistic approach to the Dirichlet problem of perturbed stable processes

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Let Xt be a symmetric stable process of index α, 0<α<2, in Rd (d≧2). In this paper we deal with the perturbation of Xt by a multiplicative functional of the following form: {Mathematical expression} with F being a function on Rd×Rd satisfying certain conditions. First we prove the following gauge theorem: If D is a bounded open domain of Rd, then the function g(x)=Ex{M(τD)} is either identically infinite on D or bounded on D, where τD is the first exit time from D. Then we formulate the Dirichlet problem associated with the perturbed symmetric stable process by using Dirichlet form theory. Finally we apply the gauge theorem to prove the existence and uniqueness of solutions to the Dirichlet problem mentioned above.

Original languageEnglish (US)
Pages (from-to)371-389
Number of pages19
JournalProbability Theory and Related Fields
Issue number3
StatePublished - Sep 1993
Externally publishedYes


  • Mathematics Subject Classification (1991): 60J30, 35S15, 60J57

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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