Appears in collection : A Random Walk in the Land of Stochastic Analysis and Numerical Probability / Une marche aléatoire dans l'analyse stochastique et les probabilités numériques

We consider a stochastic model for the evolution of a discrete population structured by a trait taking finitely many values on a grid of [0, 1], with mutation and selection. We study of the dynamics of the population in logarithm size and time scales, under a large population assumption. In the first part of the talk, individual mutations are rare but the global mutation rate tends to infinity. Then negligible sub-populations may have a strong contribution to evolution. The traits can also be horizontally transferred, leading to a trade-off between natural evolution to higher birth rates and transfer which drives the population towards lower birth rates. We prove that the stochastic discrete exponent process converges to a piecewise affine continuous function, which can be described along successive phases determined by dominant traits. In the second part of the talk, the individual mutations are small but not rare, we don't have any transfer and we assume the grid mesh for the trait values becoming smaller and smaller. We establish that under our rescaling, the stochastic discrete exponent process converges to the viscosity solution of a Hamilton-Jacobi equation, filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations. Joint works with N. Champagnat and V.C. Tran, and S. Mirrahimi for the second part.