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

Samuel Ayomide Adeosun, Peter Pfaffelhuber

TL;DR

The general limit result for martingale problems is applied to models for copy number variation of genetic elements in a diploid Moran model of size N.

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 $Φ(X^N)$ is slow for some appropriate $Φ: E\to D$. Using the method of martingale problems, we give a limit result, such that $Φ(X^N) \xRightarrow{N\to\infty} Z$ in the space of càdlàg 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 $Φ: \mathcal P(\mathbb

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

TL;DR

The general limit result for martingale problems is applied to models for copy number variation of genetic elements in a diploid Moran model of size N.

Abstract

Consider a sequence of Markov processes with state space , where has a strong drift to , such that is slow for some appropriate . Using the method of martingale problems, we give a limit result, such that in the space of càdlàg paths, and in measure. \\ We apply the general limit result to models for copy number variation of genetic elements in a diploid Moran model of size . The population by time is described by , where is the frequency of individuals with copy number , and $Φ: \mathcal P(\mathbb
Paper Structure (7 sections, 12 theorems, 77 equations)

This paper contains 7 sections, 12 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. The abstract result
  3. Modeling copy number variation of genetic elements
  4. Preparation
  5. Proof of Theorem \ref{['T2']}
  6. Case (i): Binomial/Poisson
  7. Case (ii): uniform/negative binomial

Key Result

Theorem 2.1

Let $(E,r_E)$ and $(D, r_D)$ be complete and separable spaces, $\mathcal{D}_E \subseteq \mathcal{C}_b(E)$ and $\mathcal{D}_D \subseteq \mathcal{C}_b(D)$, as well as $\Phi: E\to D$ such that $g \circ \Phi \in \mathcal{D}_E$ for $g \in \mathcal{D}_D$. Assume $X^N$ is solution of the $(G^N, \mathcal{D}

Theorems & Definitions (29)

  • Theorem 2.1
  • proof
  • Remark 3.1: Motivation for (i) and (ii)
  • Theorem 3.2
  • Remark 3.3: More general result for general $p_k$
  • Remark 3.4: $p_n = \tfrac{1}{2} \delta_n + \tfrac{1}{2} \delta_n$
  • Lemma 4.1: Dynamics on the fast time-scale
  • proof
  • Lemma 4.2: Dynamics of $\Phi(X^N)$ on the slow time-scale and limiting generator
  • proof
  • ...and 19 more