Pathwise estimates for effective dynamics: the case of nonlinear vectorial reaction coordinates

TL;DR

This paper develops pathwise estimates for effective dynamics approximations of high-dimensional diffusions, specifically addressing cases where the reaction coordinate is nonlinear or vector-valued, extending previous theoretical results.

Contribution

It provides new pathwise error estimates for effective dynamics with nonlinear or vector-valued reaction coordinates, advancing the theoretical understanding of approximation accuracy.

Findings

Pathwise estimates are derived for nonlinear reaction coordinates.

Results extend effective dynamics analysis to vector-valued reaction coordinates.

The work improves understanding of approximation errors in high-dimensional stochastic systems.

Abstract

Effective dynamics using conditional expectation was proposed in [F. Legoll and T. Leli\`evre, Nonlinearity, 2010] to approximate the essential dynamics of high-dimensional diffusion processes along a given reaction coordinate. The approximation error of the effective dynamics when it is used to approximate the behavior of the original dynamics has been considered in recent years. As a continuation of the previous work [F. Legoll, T. Leli\`evre, and S. Olla, Stoch. Process. Appl, 2017], in this paper we obtain pathwise estimates for effective dynamics when the reaction coordinate function is either nonlinear or vector-valued.