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.

We consider non-conservative positive semigroups and obtain necessary and sufficient conditions for uniform exponential contraction in weighted total variation norm. This ensures the existence of Perron eigenelements and provides quantitative estimates of spectral gaps, complementing Krein-Rutman theorems and generalizing recent results relying on probabilistic approaches. The proof is based on a non-homogenous h-transform of the semigroup and the construction of Lyapunov functions for this latter. It exploits then the classical necessary and sufficient conditions of Harris' theorem for conservative semigroups. We apply these results and obtain exponential convergence of birth and death processes conditioned on survival to their quasi-stationary distribution, as well as estimates on exponential relaxation to stationary profiles in growth-fragmentation PDEs.

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 timeinhomogeneous Markov process. We apply this technique on the example of a size-structured population with exponential growth in varying environment.Keywords: Branching Markov processes, law of large numbers, time-inhomogeneous Markov process, ergodicity.A.M.S classification: 60J80, 60F17, 60F25, 60J85, 92D25.

Abstract. We present recent results on Piecewise Deterministic Markov Processes (PDMPs), involved in biological modeling. PDMPs, first introduced in the probabilistic literature by [30], are a very general class of Markov processes and are being increasingly popular in biological applications. They also give new interesting challenges from the theoretical point of view. We give here different examples on the long time behavior of switching Markov models applied to population dynamics, on uniform sampling in general branching models applied to structured population dynamic, on time scale separation in integrate-and-fire models used in neuroscience, and, finally, on moment calculus in stochastic models of gene expression. Résumé. Nous présentons des résultats récents sur les Processus de Markov Déterministes parMorceaux (PDMPs) utilisés en modélisation en biologie. Les PDMPs, introduits pour la première fois dans la littérature probabiliste par [30], forment une classe générale de processus de Markov et sont de plus en plus populaires dans les applications en biologie. Ils fournissent également de nouveaux défis intéressant du point de vue théorique. Nous donnons ici différents exemples sur le comportement en temps long de modèles de Markov modulés appliqués à la dynamique des populations, sur le tirage uniforme dans des modèles génériques de branchement appliqués à la dynamique de populations structurées, sur les séparations d'échelles de temps dans des modèles intègre-et-tire utilisés en neuroscience, et, finalement, sur le calcul de moments dans des modèles stochastiques d'expression des gènes.