## Abstract

Let X_{t} be a symmetric stable process of index α, 0<α<2, in R^{d} (d≧2). In this paper we deal with the perturbation of X_{t} by a multiplicative functional of the following form: {Mathematical expression} with F being a function on R^{d}×R^{d} satisfying certain conditions. First we prove the following gauge theorem: If D is a bounded open domain of R^{d}, then the function g(x)=E^{x}{M(τ_{D})} is either identically infinite on D or bounded on D, where τ_{D} 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 language | English (US) |
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Pages (from-to) | 371-389 |

Number of pages | 19 |

Journal | Probability Theory and Related Fields |

Volume | 95 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1993 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty