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

¹ INSA de Rennes, 35708 Rennes, France.
² IRMAR CNRS-UMR 6625, 35000 Rennes, France.
³ Université Européenne de Bretagne, Rennes, France.
⁴ Université Grenoble Alpes, Bâtiment IMAG, 700 Avenue Centrale, 38400 Saint Martin d’Hères, France.
⁵ Université de Brest and Institut Universitaire de France, UMR CNRS 6205, Laboratoire de Mathématique de Bretagne Atlantique, 6 avenue Le Gorgeu, 29238 Brest cedex, France.

^* Corresponding author: francoise.pene@univ-brest.fr

Received: 13 April 2017
Accepted: 8 January 2019

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.