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

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