Fixation and stationary times for the $\Lambda$-Wright-Fisher process

TL;DR

This paper analyzes the fixation and stationary times of the Lambda-Wright-Fisher process with mutations, providing explicit formulas and a unified framework for understanding allele dynamics in large populations with complex reproduction patterns.

Contribution

It introduces a comprehensive approach to study fixation and stationary times in Lambda-Wright-Fisher processes, extending the fixation line to include mutation effects.

Findings

Explicit formulas for mean fixation and stationary times.

Characterization of fixation times and allele extinction order.

Extension of the fixation line to incorporate mutation effects.

Abstract

We study the fixation and stationary behavior of the Lambda-Wright-Fisher process with parent-independent mutation and finitely many types, a jump-diffusion model for allele frequency dynamics in large populations with potentially large offspring variance. Using a lookdown construction, we characterize the distribution of fixation times and the order of allele extinctions in the absence of mutations, and identify a strong stationary time in the presence of mutations. Our results include explicit expressions for the mean fixation and stationary times for the Wright-Fisher diffusion, and mean fixation times in the Beta-coalescent case. A key component of our approach is the analysis of the fixation line introduced by H\'enard in (Ann. Appl. Probab., 25:3007-3032, 2015). We extend this process to incorporate mutation, providing a unified framework for studying both fixation and equilibrium behavior.