Averaging of semigroups associated to diffusion processes on a simplex

TL;DR

This paper investigates how a diffusion process in a simplex simplifies to a jump process on the vertices when averaging over different timescales, especially under specific geometric conditions.

Contribution

It introduces a novel averaging result for diffusion processes with a degenerate noise component on a simplex, connecting continuous diffusions to jump processes.

Findings

Diffusion processes average to pure jump Markov processes on vertices.

The averaging holds under Meyer-Zheng topology.

Geometric assumptions on the simplex influence the averaging behavior.

Abstract

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.