Purely Excessive Functions and Hitting Times of Continuous-Time Branching Processes

Abstract

The aim of this note is to provide an original proof and derive fine properties of the excessive function that characterizes the Laplace transform of the downward first hitting time to a fixed level of a non-degenerate continuous-time branching process. It hinges on a recent result by Choi and Patie (2016) on the potential theory of skip-free Markov chains and reveals, in particular, that the fundamental excessive function that characterizes the first hitting time is a purely excessive function.