Long-term behavior of randomly perturbed Hamiltonian systems: large deviations and averaging

Abstract

This dissertation concerns various asymptotic problems related to the long-term macroscopic behavior of randomly perturbed Hamiltonian systems, with different types of perturbations and on different time scales. Since the perturbations of the systems are assumed to be small while the systems are observed at large times, non-trivial phenomena can arise due to the interplay between the perturbation size and the temporal and spatial scales, and the systems often demonstrate qualitatively distinct types of behavior depending on the subtle quantitative relation between asymptotic parameters.

More specifically, on a natural time scale, i.e., in time required for the dynamics to move distance of order one, we investigate the dynamical systems with fast-oscillating perturbations and obtain precise estimates on the distribution. In particular, we calculate the exact asymptotics of the distribution in the case of linear dynamical systems. This problem also inspires the study of the local limit theorem for time-inhomogeneous functions of Markov processes. The local limit theorem is a significant and widely used tool in problems of pure and applied mathematics as well as statistics. This result has been included in [1] and submitted for publication.

On the time scale that is inversely proportional to the effective size of the perturbation, we prove that the evolution of the first integral of the Hamiltonian system with fast-oscillating perturbations converges to a Markov process on the corresponding Reeb graph, with certain gluing conditions specified at the interior vertices. The result is parallel to the celebrated Freildin-Wentzell theory on the averaging principle of additive white-noise perturbations of Hamiltonian systems, and provides a description of the long-term behavior of a system when adopting an alternative approach to modeling random noise. Moreover, the current result provides the first scenario where the motion on a graph and the corresponding gluing conditions appear due to the averaging of a slow-fast system. It allows one to consider, for instance, the long-time diffusion approximation for an oscillator with a potential with more than one well. This result has been submitted for publication [2].

In the last part of the dissertation, we return to the more classical case of additive diffusion-type perturbations, combine the ideas of large deviations and averaging, and establish a large deviation principle for the first integral of the Hamiltonian system on intermediate time scales. Besides representing a new step in large deviations and averaging, this result will have important applications to reaction-diffusion equations and branching diffusions. The latter two concepts concern the evolution of various populations (e.g., in biology or chemical reactions). This result has been published in the journal Stochastics and Dynamics [3].