|Date||May 23, 2022|
Transition densities of Markov processes are of significant interest in both probability and analysis. The transition density p(t, x, y) of a Markov process with generator L is the fundamental solution of the equation partial_tu = Lu. Hence the transition density p(t, x, y) is also called as the heat kernel of L. However, an explicit expression of the heat kernel is rarely known. Due to the importance of heat kernels, there is a huge body of literature on the heat kernel estimates. The thesis consists of ve parts concerning heat kernel estimates for Markov jump processes. The rst part devotes to estimates for subordinators, namely, nondecreasing L evy processes on R. The second part studies heat kernels for non-local operators with critical killings. Then the third part considers subordinate killed Markov processes with help from the previous two parts. In particular, we give sharp estimates on the jump kernels of such processes which have degeneracy at the boundary. In the fourth part, we study heat kernel estimates for jump processes with degeneracy and critical killing via Dirichlet form theory. The last part is concerned with the fundamental solution of general time fractional equations with Dirichlet boundary condition.