Averaging principle for jump processes depending on fast ergodic dynamics

TL;DR

This paper establishes an averaging principle for slow-fast jump processes with ergodic fast dynamics, showing convergence to simpler processes and applying it to branching and epidemic models.

Contribution

It introduces a novel averaging approach for jump processes with fast ergodic dynamics and demonstrates its application to biological and epidemic models.

Findings

Convergence of the slow process to an autonomous jump process

Application to typed branching processes converging to Galton-Watson processes

Application to epidemic models converging to contact processes

Abstract

We consider a slow-fast stochastic process where the slow component is a jump process on a measurable index set whose transition rates depend on the position of the fast component. Between the jumps, the fast component evolves according to an ergodic dynamic in a state space determined by the index process. We prove that, when the ergodic dynamics are accelerated, the slow index process converges to an autonomous pure jump process on the index set. We apply our results to prove the convergence of a typed branching process toward a continuous-time Galton-Watson process, and of an epidemic model with fast viral loads dynamics to a standard contact process.