Uniform in time propagation of chaos for a Moran model

Abstract

The goal of this article is to study the limit of the empirical distribution induced by a multi-allelic Moran model, whose dynamic is given by a continuous-time irreducible Markov chain. Throughout the paper, the mutation rate driving the mutation is assumed irreducible and the selection rates are assumed uniformly bounded. The paper is divided in 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 L p over the compacts with speed of convergence of order 1/ √ N . Further on, we will consider a specific type of selection that we will call additive selection. Essentially, we assume that the selection rate can be decomposed in the sum of three elements: 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 L p of order 1/ √ N , and an asymptotic expression for the quadratic error between the empirical distribution and its limits when both, the time and 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 approximate a quasi-stationary distribution, in the same spirit of the Fleming -Viot particle systems. We then address to the problem of minimising the asymptotic quadratic error, when the time and the number of particles go to infinity. Contents 1. Introduction and main results 1 1.1. Main results 2 Structure of the paper 9 2. Links to the theory of QSD 9 2.1. Examples 10 2.2. Discussion 12 3. Proof of the main results 13 3.1. The associated martingale problem 13 3.2.

On the Stability of Positive Semigroups

“…However, to guide the reader, we end this section with some references to the literature on mean-field-type particle methodologies currently used in this context. Mean field and genetic-type particle methodologies are discussed in [52,63,64,65,66,68,69], as well as [7,27,28,126,163,181], and [3,4,5,53,54,45,97,176].…”