LAUSR

Abstract We consider a Poissonian SDE for the lack of fitness of a population subject to a continuous change of its environment, and an accumulation of advantageous mutations. We neglect the time of fixation of new mutations, so that the population is monomorphic at all times. We consider the asymptotic of small and frequent mutations. In that limit, we establish a law of large numbers and a central limit theorem. For small enough mutations, the original process is Harris recurrent and ergodic. We show in which sense the limits as t → ∞ of the law of large and number and central limit theorem give a good approximation of the invariant probability measure of the original process.