An Averaging Principle for Fast Degrees of Freedom Exhibiting Long-Term Correlations

Abstract

This article is concerned with the averaging principle and its extensions for stochastic dynamical systems with fast and slow degrees of freedom. It is demonstrated how the "conventional" averaging principle results from asymptotic multiscale analysis, how one can construct an indicator for its (in-)appropriateness, and how, if inappropriate, it may be extended into an improved approximation. The conventional scheme contains averages over the entire accessible state space of the fast degrees of freedom and may thus fail if these fast degrees of freedom exhibit long-term (auto-)correlations. In contrast, the improved scheme combines several conditional averages with a Markov jump process that is designed to represent the flipping process between the conditional averages and thus incorporates the important long-term correlations. All important steps of the derivation are illustrated by numerical experiments. Application to problems from molecular dynamics is discussed.