## Abstract

We consider a family of pseudo differential operators Δ+aΔ ^{/2}; a∈(0, 1Ş on ^{d} for every d ≥ 1 that evolves continuously from Δ to Δ+Δ^{/2}, where ∈(0, 2). It gives rise to a family of Lévy processes X^{a}, a∈(0, 1Ş in ^{d}, where X^{a} is the sum of a Brownian motion and an independent symmetric -stable process with weight a. We establish sharp two-sided estimates for the heat kernel of Δ + aΔ^{/2} with zero exterior condition in a family of open subsets, including bounded C^{1, 1} (possibly disconnected) open sets. This heat kernel is also the transition density of the sum of a Brownian motion and an independent symmetric -stable process with weight a in such open sets. Our result is the first sharp two-sided estimates for the transition density of a Markov process with both diffusion and jump components in open sets. Moreover, our result is uniform in a in the sense that the constants in the estimates are independent of a∈(0, 1Ş so that it recovers the Dirichlet heat kernel estimates for Brownian motion by taking a → 0. Integrating the heat kernel estimates in time t, we recover the two-sided sharp uniform Green function estimates of X^{a} in bounded C^{1, 1} open sets in ^{d}, which were recently established in (Z.-Q. Chen, P. Kim, R. Song and Z. Vondracek, 'Sharp Green function estimates for Δ+Δ ^{/2} in C^{1, 1} open sets and their applications', Illinois J. Math., to appear) using a completely different approach. 2011 London Mathematical Society2011

Original language | English (US) |
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Pages (from-to) | 58-80 |

Number of pages | 23 |

Journal | Journal of the London Mathematical Society |

Volume | 84 |

Issue number | 1 |

DOIs | |

State | Published - Aug 2011 |

## ASJC Scopus subject areas

- Mathematics(all)