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Probabilistic approach to the Dirichlet problem of perturbed stable processes
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  • Published: September 1993

Probabilistic approach to the Dirichlet problem of perturbed stable processes

  • Renming Song1 

Probability Theory and Related Fields volume 95, pages 371–389 (1993)Cite this article

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Summary

LetX t be a symmetric stable process of index α, 0<α<2, inR d (d≧2). In this paper we deal with the perturbation ofX t by a multiplicative functional of the following form:

$$M_t = \exp \left\{ {\sum\limits_{s\mathop \leqslant \limits_ - t} {F(X_{s - } ,X_s )} } \right\}$$

withF being a function onR d×R d satisfying certain conditions. First we prove the following gauge theorem: IfD is a bounded open domain ofR d, then the functiong(x)=E x{M(τ D )} is either identically infinite onD or bounded onD, where τ D is the first exit time fromD. 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.

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Authors and Affiliations

  1. Department of Mathematics, University of Florida, 32611, Gainesville, FL, USA

    Renming Song

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  1. Renming Song
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Song, R. Probabilistic approach to the Dirichlet problem of perturbed stable processes. Probab. Th. Rel. Fields 95, 371–389 (1993). https://doi.org/10.1007/BF01192170

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  • Received: 13 April 1992

  • Revised: 17 September 1992

  • Issue Date: September 1993

  • DOI: https://doi.org/10.1007/BF01192170

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Mathematics Subject Classification (1991)

  • 60J30
  • 35S15
  • 60J57
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