Averaging principle for diffusion processes via Dirichlet forms

TL;DR

This paper applies Dirichlet form theory to analyze diffusion processes with multi-scale drift components, providing new insights and simplified proofs for the averaging principle in finite-dimensional stochastic systems.

Contribution

It introduces a novel approach using Dirichlet forms and Mosco-convergence to study averaging principles, yielding clearer characterizations and new results for effective processes.

Findings

Simplified proofs of the averaging principle.

Characterization of the effective process via Dirichlet forms.

New results on the properties of the averaged process.

Abstract

We study diffusion processes driven by a Brownian motion with regular drift in a finite dimension setting. The drift has two components on different time scales, a fast conservative component and a slow dissipative component. Using the theory of Dirichlet form and Mosco-convergence we obtain simpler proofs, interpretations and new results of the averaging principle for such processes when we speed up the conservative component. As a result, one obtains an effective process with values in the space of connected level sets of the conserved quantities. The use of Dirichlet forms provides a simple and nice way to characterize this process and its properties.