Averaging Principle for Ordinary Differential Equations in a Multiscale Random Environment

2026/08/03 by Vincent Kagan
Mathematics · #math.PR #msc:60H10 #msc:60J25 #msc:60J76 #msc:60F17 #msc:34C29 #msc:34E13

paper · pdf

37 pages

arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

We study slow-fast stochastic systems in which a fast process, evolving within several ergodic components, drives the transition rates of a slow jump process switching between these components. Considering a differential equation whose vector field depends on both processes, we show that, under suitable ergodicity and regularity conditions, its solutions converge in law in D([0,T],\mathbb R^d) to an averaged system depending only on the slow process. The averaged drift is obtained by averaging the original vector field over the invariant measure of the fast process within each component. Our results extend classical averaging theory to systems where the vector field is explicitly influenced by a fast random environment with multi-component dynamics.

Citations

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