Dirichlet heat kernel estimates for subordinate Brownian motions with Gaussian components

Abstract

Abstract In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels, in C 1,1 open sets D in ℝ d , of a large class of subordinate Brownian motions with Gaussian components. When D is bounded, our sharp two-sided Dirichlet heat kernel estimates hold for all t > 0. Integrating the heat kernel estimates with respect to the time variable t, we obtain sharp two-sided estimates for the Green functions, in bounded C 1,1 open sets, of such subordinate Brownian motions with Gaussian components.