Dirichlet heat kernel estimates of subordinate diffusion processes with diffusive components in C1,α open sets

In this paper, we derive explicit sharp two-sided estimates of the Dirichlet heat kernel for a class of symmetric subordinate diffusion processes with diffusive components in C1,α(α∈(0,1]) open sets in Rd when the scaling order of the Laplace exponent of purely discontinuous part of the subordinator is between 0 and 1 including 1. The main result of this paper shows the stability of Dirichlet heat kernel estimates for such processes in C1,α open sets in the sense that the estimates depend on the divergence elliptic operator only via its uniform ellipticity constant and the Dini continuity modulus of the diffusion coefficients. As a corollary, we obtain sharp two-sided estimates for Green functions of those processes in bounded C1,α open sets.