On the occupation measure of evolution models with vanishing mutations

Michel Benaïm, Mario Bravo and Mathieu Faure
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Michel Benaïm: UNINE - Institut de Mathématiques - UNINE - Université de Neuchâtel = University of Neuchatel, UNINE - Université de Neuchâtel = University of Neuchatel
Mario Bravo: FAE - Facultad de Administración y Economía [Santiago de Chile]
Mathieu Faure: AMSE - Aix-Marseille Sciences Economiques - EHESS - École des hautes études en sciences sociales - AMU - Aix Marseille Université - ECM - École Centrale de Marseille - CNRS - Centre National de la Recherche Scientifique

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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 states pace, where the control on the mutation parameter is explained by the energy barrier of the limit process. Additionally, we derive an explicit L1 convergence rate, explained through the tree-optimality gap, that may be of independent interest.

Keywords: Evolution models; Inhomogeneous Markov chain; Occupation measure; Energy barrier (search for similar items in EconPapers)
Date: 2026-02
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