Markov Processes along Continuous Time Galton-Watson Trees

Abstract

Let us note that an individual may die without descendance when p ₀ > 0. Moreover, when X is a Feller diffusion and p ₂ = 1, we recover the splitting Feller diffusion of Chapter 8 In the general case, the process X is no longer a branching process and the key property for the long time study of the measure-valued process will be the ergodicity of a well-chosen auxiliary Markov process. A vast literature can be found concerning branching Markov processes and special attention has been payed to Branching Brownian Motion from the pioneering work of Biggins [10] about branching random walks, see, e.g., [28, 60] and the references therein. More recently, non-local branching events (with jumps occurring at the branching times) and superprocesses limits corresponding to small and rapidly branching particles have been considered and we refer, e.g., to the works of Dawson et al. and Dynkin [26].