Harmonic Functions and the Dirichlet Problem for Revival Markov Processes

Abstract

Harmonic functions and the Dirichlet problem on an open set $G$ are defined for a pieced-out or revival Markov process constructed from a continuous base process. A one-to-one correspondence is obtained between the bounded harmonic functions of the revival process and those of the base process. The harmonic functions of the revival process are shown to be continuous on $G$ under certain conditions and to coincide with the solutions of $\mathscr{U}f = 0$ on $G$ where $\mathscr{U}$ is the characteristic operator. These results are applied to random evolution processes and branching processes.