Abstract:
We analyze the long-term behavior of evolutionary models where mutations gradually vanish over time. Our focus is on the almost sure convergence of the empirical occupation measure—that is, the time-averaged distribution of states—as the mutation parameter decays in a controlled manner. Under suitable conditions, we prove that this measure converges almost surely to a specific invariant distribution of a limiting Markov chain, while also establishing explicit convergence rates. Our analysis is carried out within a class of time-inhomogeneous Markov chains on a finite state space, where the so-called energy barrier of the limiting dynamics determines the required decay rate of the mutation parameter.