Dirichlet heat kernel estimates of subordinate diffusion processes with diffusive components in $C^{1, \alpha}$ open sets

Abstract

In this paper, we derive explicit sharp two-sided estimates of the Dirichlet heat kernels for a class of symmetric subordinate diffusion processes with diffusive components in $C^{1, \alpha}(\alpha\in (0, 1])$ open sets in $\mathbb R^d$ 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 $C^{1, \alpha}$ 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 the sharp two-sided estimates for Green functions of those processes in bounded $C^{1, \alpha}$ open sets.