  We prove the existence and uniqueness of a quasi-stationary distribution for
three stochastic processes derived from the model of Muller's ratchet. This
model was invented with the aim of evaluating the limitations of an asexual
reproduction mode in preventing the accumulation of deleterious mutations
through natural selection alone. The main considered model is non-classical, as
it is a stochastic diffusion evolving on an irregular set of infinite dimension
with hard killing on an hyperplane. We are nonetheless able to prove
exponential convergence in total variation to the quasi-stationary distribution
even in this case. The parameters in this last convergence result are directly
related to the core parameters of Muller's ratchet. The speed of convergence to
the quasi-stationary distribution is deduced both for the infinite dimensional
model and for approximations with a large yet finite number of potential
mutations. Likewise, we give uniform moment estimates of the empirical
distribution of mutations in the population under quasi-stationarity.
