  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.
