Heat kernel estimates for Δ+Δ α /2 in C 1, 1 open sets

Abstract

We consider a family of pseudo differential operators {Δ+aα Δα/2; a∈(0, 1]} on ℝd for every d ⩾ 1 that evolves continuously from Δ to Δ+Δα/2, where α∈(0, 2). It gives rise to a family of Lévy processes {Xa, a∈(0, 1]} in ℝd, where Xa is the sum of a Brownian motion and an independent symmetric α-stable process with weight a. We establish sharp two-sided estimates for the heat kernel of Δ + aα Δα/2 with zero exterior condition in a family of open subsets, including bounded C1, 1 (possibly disconnected) open sets. This heat kernel is also the transition density of the sum of a Brownian motion and an independent symmetric α-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, 1] 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 Xa in bounded C1, 1 open sets in ℝd, which were recently established in (Z.-Q. Chen, P. Kim, R. Song and Z. Vondracek, ‘Sharp Green function estimates for Δ+Δα/2 in C1, 1 open sets and their applications’, Illinois J. Math., to appear) using a completely different approach.