Limit theorems for some branching measure-valued processes
Résumé
We consider a particles system, where, the particles move independently according to a Markov process and branching event occurs at an inhomogeneous time. The offspring locations and their number may depend on the position of the mother. Our setting capture, for instance, the processes indexed by Galton-Watson tree. We first determine the asymptotic behaviour of the empirical measure. The proof is based on an expression of the empirical measure using an auxiliary process. This latter is not distributed as a one cell lineage, there is a biased phenomenon. Our model is a microscopic description of a random (discrete) population of individuals. We then obtain a large population approximation as weak solution of a growth- fragmentation equation. We illustrate our result with two examples. The first one is a size-structured population model which describes the mitosis and the second one can model a parasite infection.
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