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

TL;DR

This paper establishes a limit theorem for a sequence of Markov processes with strong drift towards a subspace, and applies it to genetic models describing copy number variation in populations.

Contribution

It introduces a martingale problem approach to analyze the asymptotic behavior of Markov processes with large drift, with applications to population genetics models.

Findings

Convergence of projected processes to a limiting process Z

Convergence of the full process X^N to X in measure

Application to genetic copy number variation models

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 $\Phi(X^N) \xRightarrow{N\to\infty} Z$ in the space of c\`adl\`ag paths, and $X^N \xRightarrow{N\to\infty} X$ 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 $X^N_k$ is the frequency of individuals with copy number $k$, and $\Phi: \mathcal P(\mathbb