Heat Kernel Estimate for ∆+∆ in C open sets

We consider a family of pseudo differential operators {∆ + a∆; a ∈ (0, 1]} on R for every d ≥ 1 that evolves continuously from ∆ to ∆ + ∆, where α ∈ (0, 2). It gives rise to a family of Lévy processes {X, a ∈ (0, 1]} in R, where X 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 ∆ + a∆ with zero exterior condition in a family of open subsets, including bounded C (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 X in bounded C open sets in R, which were recently established in [14] by using a completely different approach. AMS 2000 Mathematics Subject Classification: Primary 60J35, 47G20, 60J75; Secondary 47D07