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

Let $Z=(Z_t{)}_{t\in [0,\mathrm{\infty })}$ be an ergodic Markov process and, for every $n\in \mathbb{N}$, let $Z^n=(Z_{n^{2}t}{)}_{t\in [0,\mathrm{\infty })}$ drive a process $X^n$. Classical results show under suitable conditions that the sequence of non-Markovian processes $(X^n{)}_{n\in \mathbb{N}}$ converges to a Markov process and give its infinitesimal characteristics. Here, we consider a general sequence $(Z^n{)}_{n\in \mathbb{N}}$. Using a general result on stochastic averaging 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 fluctuating selection.