Large Deviations for Hamiltonian Systems on Intermediate Time Scales

TL;DR

This paper investigates large deviations for a perturbed Hamiltonian system on intermediate time scales, establishing a large deviation principle with an action functional linked to the averaged process on the Reeb graph.

Contribution

It extends large deviation analysis to intermediate time scales for Hamiltonian systems with state-dependent noise, bridging finite and long-term behaviors.

Findings

Proves large deviation principle on intermediate time scales.

Derives the action functional based on the averaged process.

Shows convergence of the projected process to a Markov process.

Abstract

We consider a two-dimensional Hamiltonian system perturbed by a small diffusion term, whose coefficient is state-dependent and non-degenerate. As a result, the process consists of the fast motion along the level curves and slow motion across them. On finite time intervals, the large deviation principle applies, while on time scales that are inversely proportional to the size of the perturbation, the averaging principle holds, i.e., the projection of the process onto the Reeb graph converges to a Markov process. In our paper, we consider the intermediate time scales and prove the large deviation principle, with the action functional determined in terms of the averaged process on the graph.