Averaging of semigroups associated to diffusion processes on a simplex

We study the averaging of a diffusion process living in a simplex $K$ of ${\mathbb{R}}^n$, $n\ge 1$. We assume that its infinitesimal generator can be decomposed as a sum of two generators corresponding to two distinct timescales and that the one corresponding to the fastest timescale is pure noise with a diffusion coefficient vanishing exactly on the vertices of $K$. We show that this diffusion process averages to a pure jump Markov process living on the vertices of $K$ for the Meyer-Zheng topology. The role of the geometric assumptions done on $K$ is also discussed.