Multitype Galton-Watson processes in random environment and products of random operators
par
Amphi Schwartz
Galton-Watson processes form a large class of discrete time population models, in which individuals engage in an asexual and random reproduction, without interacting with each other. In this talk, we study processes of this class in which each individual has a type and the reproduction of individuals is affected by a changing environment represented by a stationary and ergodic process. Namely, the probability distribution of the offpsring of an individual depends both on its type and the state of the environment at the time he lives. The study of such a process relies on the understanding of its quenched mean, that is, the mean of the population conditionally on the environmental process. This mean is linked to a product of random positive linear operators which act on some measures and functions space. In the case where there is only a finite number of possible types, these operators are merely positive matrices. The rich theory of products of random matrices, intiated in the 1960s, has allowed to obtain precise results on the associated Galton-Watson processes in the last decades. We focus here on the case of an infinite set of possible types. We start by obtaining a ergodicity result for the associated products of operators, from which we deduce sufficient condition for almost sure extinction, as well as a description of the surviving population when there is one, under the form of a Kesten-Stigum-type theorem.