Uniform in time propagation of chaos for a Moran model

The goal of this article is to study the limit of the empirical distribution induced by a mutation-selection multi-allelic Moran model, whose dynamic is given by a continuous-time irreducible Markov chain. The rate matrix driving the mutation is assumed irreducible and the selection rates are assumed uniformly bounded. The paper is divided into two parts. The first one deals with processes with general selection rates. For this case we are able to prove the propagation of chaos in $\mathbb{L}^p$ over the compacts, with speed of convergence of order $1/\sqrt{N}$. Further on, we consider a specific type of selection that we call additive selection. Essentially, we assume that the selection rate can be decomposed as the sum of three terms: a term depending on the allelic type of the parent (which can be understood as selection at death), another term depending on the allelic type of the descendant (which can be understood as selection at birth) and a third term which is symmetric. Under this setting, our results include a uniform in time bound for the propagation on chaos in $\mathbb{L}^p$ of order $1/\sqrt{N}$, and the proof of the asymptotic normality with zero mean and explicit variance, for the approximation error between the empirical distribution and its limit, when the number of individuals tend towards infinity. Additionally, we explore the interpretation of the Moran model with additive selection as a particle process whose empirical distribution approximates a quasi-stationary distribution, in the same spirit as the Fleming\,--\,Viot particle systems. We then address the problem of minimising the asymptotic quadratic error, when the time and the number of particles go to infinity.