Time Reversal of Some Stationary Jump-Diffusion Processes from Population Genetics

TL;DR

This paper characterizes the time reversal of stationary jump-diffusion processes modeling genetic variation, revealing how boundary accessibility influences the reversed process's behavior, with implications for population genetics modeling.

Contribution

It provides a detailed analysis of the time-reversal dynamics of jump-diffusions in population genetics, highlighting boundary accessibility effects.

Findings

Time reversal depends on boundary accessibility.

Reversed process jumps immediately into interior at inaccessible boundaries.

Jumps at accessible boundaries are governed by weighted local time.

Abstract

We describe the processes obtained by time reversal of a class of stationary jump-diffusion processes that model the dynamics of genetic variation in populations subject to repeated bottlenecks. Assuming that only one lineage survives each bottleneck, the forward process is a diffusion on [0,1] that jumps to the boundary before diffusing back into the interior. We show that the behavior of the time-reversed process depends on whether the boundaries are accessible to the diffusive motion of the forward process. If a boundary point is inaccessible to the forward diffusion, then time reversal leads to a jump-diffusion that jumps immediately into the interior whenever it arrives at that point. If, instead, a boundary point is accessible, then the jumps off of that point are governed by a weighted local time of the time-reversed process.