On the occupation measure of evolution models with vanishing mutations
Michel Benaïm, Mario Bravo, Mathieu Faure
TL;DR
This work analyzes the long-term behavior of occupation measures in evolution models when mutation noise vanishes over time. It builds a time-inhomogeneous Markov framework from a homogeneous rare-transition model and characterizes convergence to the invariant distribution $\pi^*$, governed by the energy barrier $e(\mathbf{c})$ and the tree-optimality gap $\theta$. The main result proves almost-sure convergence of the empirical occupation measure $v_n$ to $\pi^*$ under a slow decay condition $2A e(\mathbf{c})<1$, and provides explicit $L^1$ rates that depend on the barrier structure and coradius-like quantities, with an optimal rate achieved at $A_{\mathrm{crit}}$. The analysis fuses tree-based potential methods, spectral-gap bounds for nonreversible chains, and stochastic-approximation techniques to derive both convergence and quantitative rates, offering insights into stochastic stability under vanishing exploration in evolutionary settings.
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 $L^1$ convergence rate, explained through the tree-optimality gap, that may be of independent interest.