On recurrent properties and convergence rates of generalised Fisher -- Wright's diffusion with mutation

TL;DR

The paper analyzes a generalized one-dimensional Fisher–Wright diffusion with mutations, modeled by $dX_t = B(X_t) dt + \epsilon \sigma(X_t) dW_t$ on $(0,1)$ with drift bounds and a nondegenerate diffusion term. The authors develop a framework based on Lyapunov functions and a 1D intersection-time local mixing argument to prove exponential ergodicity under minimal coefficient regularity. They establish the existence of a finite invariant measure $\mu$ via a Khasminskii-type construction and prove finiteness of $\int_0^1 (x^{-m}+(1-x)^{-m})\,\mu(dx)$ for suitable $m$, and furthermore show exponential convergence in total variation to $\mu$ with explicit rate bounds and uniqueness. These results extend previous Wright–Fisher analyses by relaxing regularity assumptions and providing explicit exponential convergence rates under a one-dimensional, coupling-free approach.

Abstract

A generalised one-dimensional Fisher-Wright diffusion process with mutations is considered. This is a well-known model in population genetics. An exponential recurrence is established for the process, which also implies an exponential rate of convergence towards the invariant measure.

Theorems & Definitions (14)

• Lemma 1
• proof
• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 1
• proof
• Corollary 1
• proof