Volume 23, 2019
|Page(s)||638 - 661|
|Published online||26 September 2019|
A law of large numbers for branching Markov processes by the ergodicity of ancestral lineages
Univ. Grenoble Alpes, INRIA,
* Corresponding author: firstname.lastname@example.org
Accepted: 20 December 2018
We are interested in the dynamic of a structured branching population where the trait of each individual moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event occurs, the trait of a descendant at birth depends on the trait of the mother. We prove a law of large numbers for the empirical distribution of ancestral trajectories. It ensures that the empirical measure converges to the mean value of the spine which is a time-inhomogeneous Markov process describing the trait of a typical individual along its ancestral lineage. Our approach relies on ergodicity arguments for this time-inhomogeneous Markov process. We apply this technique on the example of a size-structured population with exponential growth in varying environment.
Mathematics Subject Classification: 60J80 / 60F17 / 60F25 / 60J85 / 92D25
Key words: Branching Markov processes / law of large numbers / time-inhomogeneous Markov process / ergodicity
© EDP Sciences, SMAI 2019
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