Large deviations and adiabatic transitions for dynamical systems and Markov processes in fully coupled averaging

TL;DR

This paper investigates large deviations and transition phenomena in coupled slow-fast dynamical systems and Markov processes, providing insights into their long-term behavior and rare event probabilities.

Contribution

It introduces a unified approach to analyze large deviations and adiabatic transitions in systems with fully coupled averaging, covering hyperbolic dynamical systems and Markov processes.

Findings

Derived large deviation principles for slow motions

Characterized transition times between attractors

Extended results to systems with combined diffusions and Markov chains

Abstract

The work treats systems combining slow and fast motions depending on each other where fast motions are perturbations of families of either dynamical systems or Markov processes with freezed slow variable. In the first case we consider hyperbolic dynamical systems and in the second case we deal with random evolutions which are combinations of diffusions and continuous time Markov chains. We study first large deviations of the slow motion from the averaged one and then use these results together with some Markov property type arguments in order to describe very long time behavior of the slow motion such as its transitions between attractors of the averaged system.