On the occupation measure of evolution models with vanishing mutations

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.

💡 Research Summary

**
The paper investigates the long‑run behavior of the empirical occupation measure generated by a class of time‑inhomogeneous Markov chains that arise in evolutionary game‑theoretic models where mutation (or experimentation) probabilities decay over time. The authors start by formalizing a homogeneous evolution model: a finite state space $S$, an admissible cost function $c:S\times S\to