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We prove the existence and uniqueness of a quasi-stationary distribution forthree stochastic processes derived from the model of the M\"uller ratchet. Thismodel has been originally introduced to quantify the limitations of a purelyasexual mode of reproduction in preventing, only through natural selection, thefixation and accumulation of deleterious mutations. The main considered modelis clearly non-classical, as it is a stochastic diffusion evolving on anirregular set of infinite dimension with hard killing on an hyperplane. We arenonetheless able to prove exponential convergence in total variation to thequasi-stationary distribution even in this case. The parameters in this lastresult of convergence are directly related to the core parameters of theM\"uller ratchet effect. The speed of convergence to the quasi-stationarydistribution deduced from the infinite dimensional model extends to theapproximations with a large yet finite number of potential mutations. Likewise,we have uniform moment estimates of the empirical distribution of mutations inthe population under quasi-stationarity.

Metastability between the clicks of the M\"uller ratchet