Coarse-graining of non-reversible stochastic differential equations: quantitative results and connections to averaging

TL;DR

This paper develops quantitative error estimates for model reduction of non-reversible stochastic differential equations by coarse-graining, extending previous reversible results and analyzing the relationship between effective and averaged dynamics.

Contribution

It extends effective reversible dynamics results to non-reversible processes with non-constant diffusion, providing error bounds and insights into their relation to averaging.

Findings

Relative entropy and Wasserstein error bounds established.

Effective dynamics may differ from averaged equations in non-reversible systems.

Numerical examples illustrate theoretical error estimates.

Abstract

This work is concerned with model reduction of stochastic differential equations and builds on the idea of replacing drift and noise coefficients of preselected relevant, e.g. slow variables by their conditional expectations. We extend recent results by Legoll & Leli\`evre [Nonlinearity 23, 2131, 2010] and Duong et al. [Nonlinearity 31, 4517, 2018] on effective reversible dynamics by conditional expectations to the setting of general non-reversible processes with non-constant diffusion coefficient. We prove relative entropy and Wasserstein error estimates for the difference between the time marginals of the effective and original dynamics as well as an entropy error bound for the corresponding path space measures. A comparison with the averaging principle for systems with time-scale separation reveals that, unlike in the reversible setting, the effective dynamics for a non-reversible system need not agree with the averaged equations. We present a thorough comparison for the Ornstein-Uhlenbeck process and make a conjecture about necessary and sufficient conditions for when averaged and effective dynamics agree for nonlinear non-reversible processes. The theoretical results are illustrated with suitable numerical examples.