A probabilistic analysis of a continuous-time evolution in recombination

TL;DR

This paper provides a probabilistic analysis of the continuous-time recombination equation in population genetics, describing the evolution as a Markov chain on partitions, with explicit law, decay rates, and quasi-stationary behavior.

Contribution

It introduces a novel probabilistic framework using trees to analyze the evolution of recombination, including explicit law descriptions and decay rate analysis.

Findings

Markov chain on partitions converges to the finest partition

Explicit law of the process using a family of trees

Identifies geometric decay rate and quasi-stationary behavior

Abstract

We study the continuous-time evolution of the recombination equation of population genetics. This evolution is given by a differential equation that acts on a product probability space, and its solution can be described by a Markov chain on a set of partitions that converges to the finest partition. We study an explicit form of the law of this process by using a family of trees. We also describe the geometric decay rate to the finest partition and the quasi-stationary behavior of the Markov chain when conditioned on the event that the chain does not hit the limit.