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Let $Z = (Z_t)_{t\in[0,\infty)}$ be an ergodic Markov process and, for every$n\in\mathbb{N}$, let $Z^n = (Z_{n^2 t})_{t\in[0,\infty)}$ drive a process$X^n$. Classical results show under suitable conditions that the sequence ofnon-Markovian processes $(X^n)_{n\in\mathbb{N}}$ converges to a Markov processand give its infinitesimal characteristics. Here, we consider a generalsequence $(Z^n)_{n\in\mathbb{N}}$. Using a general result on stochasticaveraging from [Kur92], we derive conditions which ensure that the sequence$(X^n)_{n\in\mathbb{N}}$ converges as in the classical case. As an application,we consider the diffusion limit of a Wright-Fisher model with fluctuatingselection.

Stochastic averaging for multiscale Markov processes with an application to a Wright-Fisher model with fluctuating selection