Wright-Fisher diffusion with negative mutation rates

TL;DR

This paper introduces a novel family of Wright-Fisher diffusions with nonpositive mutation rates, analyzing their boundary behavior, exit distributions, and connection to time-reversed processes using Bessel-square diffusions of negative dimensions.

Contribution

It extends Wright-Fisher models to negative mutation rates, providing explicit exit distributions and a stochastic time-reversal framework using Bessel-square processes.

Findings

Explicit boundary exit distribution derived

Probabilistic bounds on exit times established

Processes linked to time-reversal of classical Wright-Fisher models

Abstract

We study a family of n-dimensional diffusions, taking values in the unit simplex of vectors with nonnegative coordinates that add up to one. These processes satisfy stochastic differential equations which are similar to the ones for the classical Wright-Fisher diffusions, except that the "mutation rates" are now nonpositive. This model, suggested by Aldous, appears in the study of a conjectured diffusion limit for a Markov chain on Cladograms. The striking feature of these models is that the boundary is not reflecting, and we kill the process once it hits the boundary. We derive the explicit exit distribution from the simplex and probabilistic bounds on the exit time. We also prove that these processes can be viewed as a "stochastic time-reversal" of a Wright-Fisher process of increasing dimensions and conditioned at a random time. A key idea in our proofs is a skew-product construction using certain one-dimensional diffusions called Bessel-square processes of negative dimensions, which have been recently introduced by Going-Jaeschke and Yor.