Conditioning and limit theorems for random walks and branching processes – RAWABRANCH

This project focuses on various aspects of branching processes in fixed, variable or random environments, whether they are single-type or multitype. We propose to identify the limit of Bienaymé-Galton-Watson trees conditioned by their total population through their coding by multi-indexed and matrix-valued random walks. Then we will study the problem of the extinction of a part of the population for continuous multitype branching processes. We will construct the continuous analogue of multitype Bienaymé-Galton-Watson trees. These continuous random trees will then be obtained in the stable case as scaling limits of the renormalized discrete trees. These continuous random trees will be associated with continuous multi-type branching processes. We will also study discrete-time multitype branching processes in random environments to obtain asymptotic properties of the corresponding population size and survival probability; in particular, the problems of large deviations and asymptotic normalization will be considered. To this end, we will first deepen the study of the products of random matrices, in particular through the study of the multidimensional processes corresponding to the linear action of these products of matrices. We will be particularly interested in the cases where these processes are conditioned to remain in a cone of the Euclidean space. We will then establish limit theorems (invariance principle, local limit theorem, ...) for these conditioned processes. We will finally focus on the fundamental branching martingale associated to these Bienaymé-Galton-Watson trees, defined from the corresponding products of random matrices.