Exponential growth of branching processes in a general context of lifetimes and birthtimes dependence

Abstract

We study the exponential growth of branching processes with ancestral dependence. We suppose here that the lifetimes of the cells are dependent random variables, that the numbers of new cells are random and dependent. Lifetimes and new cells’s numbers are also assumed to be dependent. Applying the spectral study of Laplace-type operators recently made in Hervé et al. [ESAIM: PS 23 (2019) 607–637], we illustrate our results in the Markov context, for which the exponential growth property is linked to the Laplace transform of the lifetimes of the cells.