We consider an infinite-dimensional system of stochastic differential equations describing the evolution of type frequencies in a large population. The type of an individual is the number of deleterious mutations it carries, where fitness of individuals carrying $k$ mutations is decreased by $\alpha k$ for some $\alpha>0$. Along the individual lines of descent, new mutations accumulate at rate $\lambda$ per generation, and each of these mutations has a probability $\gamma$ per generation to disappear. While the case $\gamma=0$ is known as (the Fleming–Viot version of) Muller’s ratchet, the case $\gamma>0$ is associated with compensatory mutations in the biological literature. We show that the system has a unique weak solution. In the absence of random fluctuations in type frequencies (i.e., for the so-called infinite population limit) we obtain the solution in a closed form by analyzing a probabilistic particle system and show that for $\gamma>0$, the unique equilibrium state is the Poisson distribution with parameter $\lambda/(\gamma+\alpha)$.
"Muller’s ratchet with compensatory mutations." Ann. Appl. Probab. 22 (5) 2108 - 2132, October 2012. https://doi.org/10.1214/11-AAP836