Uniform sampling in a structured branching population

Abstract

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 the descendants at birth depends on the trait of the mother and on the number of descendants. In this article, we explicitly describe the penalized Markov process, named auxiliary process, corresponding to the dynamic of the trait of a "typical" individual by giving its associated infinitesimal generator. We prove a Many-to-One formula and a Many-to-One formula for forks. Furthermore, we prove that this auxiliary process characterizes exactly the process of the trait of a uniformly sampled individual in a large population approximation. We detail three examples of growth-fragmentation models: the linear growth model, the exponential growth model and the parasite infection model.

Ergodic Behavior of Non-conservative Semigroups via Generalized Doeblin’s Conditions

“…for which ergodic behaviour can be obtained through coupling arguments. This auxiliary semigroup describes the trajectory of a typical particle and has been used recently for the study of branching Markov processes in discrete and continuous time [4,5,2,34] and processes killed at a boundary [11,18,35]. We come back in Appendix A on the link between these topics in probability and ergodic estimates for semigroups.…”

Statistical estimation in a randomly structured branching population

“…Under Assumptions 2 and 3, the process X in (2) is well defined and the size of the population does not explode in finite time almost-surely, see for instance Marguet [28]. Note that the lower bounds for σ and B are not needed for the well-posedness of X but rather for later statistical purposes.…”

“…where F (t, x) is the cumulative density function of W x (t). Therefore, combining (28) and (29) we get…”

“…In many cases, this approach is linked to certain piecewise deterministic Markov models or bifurcating Markov chains (BMC) in discrete time. These models are well understood from a probabilist point of view (in discrete time Guyon [19], Bitseki-Penda et al [8,9], in continuous time Bansaye and Méléard [4], Bansaye et al [3] or more recently Marguet [28] for a general approach). For the statistical estimation, we refer to [10,16,17,23,5], and the references therein, see also Bitseki-Penda and Olivier [31], de Saporta et al [14,15], Azaïs et al [1] or recently Bitseki-Penda and Roche [7].…”

A law of large numbers for branching Markov processes by the ergodicity of ancestral lineages

“…In this article, we prove the convergence of the empirical measure for a class of general branching Markov processes, using spinal techniques. More precisely, we use the characterization of the trait along a typical ancestral lineage introduced in [Mar16]. We adapt the techniques of [HM11] and we prove that under classical conditions [MT12,Chapters 15,16], the semigroup of the auxiliary process, which is a time-inhomogeneous Markov process, is ergodic.…”