Global Heat Kernel Estimates for Relativistic Stable Processes in Half-space-like Open Sets

Abstract

In this paper, by using probabilistic methods, we establish sharp two-sided large time estimates for the transition densities of relativistic α-stable processes with mass m ∈ (0, 1] (i.e., for the Dirichlet heat kernels of m − (m 2/α − Δ) α/2 with m ∈ (0, 1]) in half-space-like C 1, 1 open sets. The estimates are uniform in m in the sense that the constants are independent of m ∈ (0, 1]. Combining with the sharp two-sided small time estimates, established in Chen et al. (Ann Probab, 2011), valid for all C 1, 1 open sets, we have now sharp two-sided estimates for the transition densities of relativistic α-stable processes with mass m ∈ (0, 1] in half-space-like C 1, 1 open sets for all times. Integrating the heat kernel estimates with respect to the time variable, one can recover the sharp two-sided Green function estimates for relativistic α-stable processes with mass m ∈ (0, 1] in half-space-like C 1, 1 open sets established recently in Chen et al. (Stoch Process their Appl, 2011).