In this paper, we establish sharp two-sided estimates for the transition densities of relativistic stable processes [i.e., for the heat kernels of the operators m − (m2/α − Δ)α/2] in C1,1 open sets. Here m > 0 and α ∈ (0, 2). The estimates are uniform in m ∈ (0, M] for each fixed M > 0. Letting m ↓ 0, we recover the Dirichlet heat kernel estimates for Δα/2 := −(−Δ)α/2 in C1,1 open sets obtained in . Sharp two-sided estimates are also obtained for Green functions of relativistic stable processes in bounded C1,1 open sets.
"Sharp heat kernel estimates for relativistic stable processes in open sets." Ann. Probab. 40 (1) 213 - 244, January 2012. https://doi.org/10.1214/10-AOP611