Coarse Graining of Nonreversible Stochastic Differential Equations: Quantitative Results and Connections to Averaging

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 and Leli\` evre [Nonlinearity 23 (2010), pp. 2131--2165] and Duong et al. [Nonlinearity, 31 (2018), pp. 4517--4567] on effective reversible dynamics by conditional expectations to the setting of general nonreversible processes with nonconstant 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 nonreversible 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 nonreversible processes. The theoretical results are illustrated with suitable numerical examples.