On the occupation measure of evolution models with vanishing mutations

TL;DR

This paper proves the almost sure convergence of occupation measures in evolution models with decreasing mutation rates, linking the rate to the invariant distribution of a limiting Markov chain.

Contribution

It establishes convergence results for occupation measures under controlled mutation decay and provides explicit convergence rates related to the energy barrier.

Findings

Occupation measure converges almost surely to a specific invariant distribution.

Convergence rate is explicitly derived and related to the tree-optimality gap.

Results apply to a class of time-inhomogeneous Markov chains with finite states.

Abstract

We study the almost sure convergence of the occupation measure of evolution models where mutation rates decrease over time. We show that if the mutation parameter vanishes at a controlled rate, then the empirical occupation measure converges almost surely to a specific invariant distribution of a limiting Markov chain. Our results are obtained through the analysis of a larger class of time-inhomogeneous Markov chains with finite state space, where the control on the mutation parameter is explained by the energy barrier of the limit process. Additionally, we derive an explicit convergence rate, explained through the tree-optimality gap, that may be of independent interest.