Markov processes forced on a subspace by a large drift, with applications to population genetics

Consider a sequence of Markov processes $X^1,X^2,\mathrm{.}\mathrm{.}\mathrm{.}$ with state space $E$, where $X^N$ has a strong drift to $D\subseteq E$, such that $\mathrm{\Phi }(X^N)$ is slow for some appropriate $\mathrm{\Phi }:E\to D$. Using the method of martingale problems, we give a limit result, such that $\mathrm{\Phi }(X^N)\stackrel{N\to \mathrm{\infty }}{\Rightarrow }Z$ in the space of càdlàg paths, and $X^N\stackrel{N\to \mathrm{\infty }}{\Rightarrow }X$ in measure. \ We apply the general limit result to models for copy number variation of genetic elements in a diploid Moran model of size $N$. The population by time $t$ is described by $X^N\in \mathcal{P}({\mathbb{N}}_0)$, where $X_k^N$ is the frequency of individuals with copy number $k$, and $Φ: \mathcal P(\mathbb