Lévy Processes and the Genealogy of General Continuous-state Branching Processes

Abstract

The Brownian snake construction of quadratic superprocesses relies on the fact that the genealogical structure of the Feller diffusion can be coded by reflected Brownian motion. Our goal in this chapter is to explain a similar coding for the genealogy of continuous-state branching processes with a general branching mechanism ψ. The role of reflected Brownian motion will be played by a certain functional of a Levy process with no negative jumps and Laplace exponent ψ. We first explain the key underlying ideas in a discrete setting.