Scaling limits of population and evolution processes in random environment

Abstract

We propose a general method for investigating scaling limits of finite dimensional Markov chains to diffusions with jumps. The results of tightness, identification and convergence in law are based on the convergence of suitable characteristics of the chain transition. We apply these results to population processes recursively defined as sums of independent random variables. Two main applications are developed. First, we extend the Wright-Fisher model to independent and identically distributed random environments and show its convergence, under a large population assumption, to a Wright-Fisher diffusion in random environment. Second, we obtain the convergence in law of generalized Galton-Watson processes with interaction in random environment to solutions of stochastic differential equations with jumps.