## Abstract

In this paper, we consider a large class of purely discontinuous rotationally symmetric Ĺevy processes. We establish sharp two-sided estimates for the transition densities of such processes killed upon leaving an open set D. When D is a ?-fat open set, the sharp two-sided estimates are given in terms of surviving probabilities and the global transition density of the Ĺevy process. When D is a C1,1 open set and the Ĺevy exponent of the process is given by ?(?) = f(|?|2) with f being a complete Bernstein function satisfying a mild growth condition at infinity, our two-sided estimates are explicit in terms of ?, the distance function to the boundary of D and the Ĺevy density of X. This gives an affirmative answer to the conjecture posted in Chen, Kim and Song [Global heat kernel estimates for relativistic stable processes in half-space-like open sets. Potential Anal. 36 (2012) 235-261]. Our results are the first sharp two-sided Dirichlet heat kernel estimates for a large class of symmetric Ĺevy processes with general Ĺevy exponents. We also derive an explicit lower bound estimate for symmetric Ĺevy processes on Rd in terms of their Ĺevy exponents.

Original language | English (US) |
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Pages (from-to) | 90-120 |

Number of pages | 31 |

Journal | Proceedings of the London Mathematical Society |

Volume | 109 |

Issue number | 1 |

DOIs | |

State | Published - Jul 2014 |

## ASJC Scopus subject areas

- General Mathematics