Markov processes forced on a subspace by a large drift, with applications to population genetics

Abstract

Consider a sequence of Markov processes $X^1,X^2,...$ with state space $E$, where $X^N$ has a strong drift to $D\subseteq E$, such that $\Phi (X^N)$ is slow for some appropriate $\Phi :E\to D$. Using the method of martingale problems, we give a limit result, such that $\Phi (X^N)\stackrel{N\to \infty }{\Rightarrow }Z$ in the space of càdlàg paths, and $X^N\stackrel{N\to \infty }{\Rightarrow }X$ in measure. \\ We apply the general limit result to models for copy number variation of genetic elements in a diploid Moran model of size $N$. The population by time $t$ is described by $X^N\in \mathcal{P}({\mathbb{N}}_0)$, where $X_k^N$ is the frequency of individuals with copy number $k$, and $Φ: \mathcal P(\mathbb

17 pages