A non-standard evolution problem arising in population genetics

TL;DR

This paper investigates a specialized Fokker-Planck equation modeling population genetics, establishing existence, uniqueness, and convergence of solutions with a focus on boundary behaviors and singular measures.

Contribution

It introduces a novel weak solution framework for a degenerate Fokker-Planck equation with boundary conservation laws, proving key properties including convergence to a singular stationary measure.

Findings

Existence and uniqueness of weak solutions are established.

Solutions converge exponentially to a singular stationary measure.

The stationary solution is supported at the boundaries, representing population fixation.

Abstract

We study the evolution of the probability density of an asexual, one locus population under natural selection and random evolution. This evolution is governed by a Fokker-Planck equation with degenerate coefficients on the boundaries, supplemented by a pair of conservation laws. It is readily shown that no classical or standard weak solution definition yields solvability of the problem. We provide an appropriate definition of weak solution for the problem, for which we show existence and uniqueness. The solution displays a very distinctive structure and, for large time, we show convergence to a unique stationary solution that turns out to be a singular measure supported at the endpoints. An exponential rate of convergence to this steady state is also proved.