Most Probable KAM Tori in Stochastic Hamiltonian Systems Driven by Multiplicative Noise

TL;DR

This paper extends KAM theory to stochastic Hamiltonian systems with multiplicative noise, showing the persistence of invariant tori and characterizing deviation probabilities using large deviation principles.

Contribution

It proves the persistence of most probable invariant tori under stochastic perturbations and derives the rate function for deviations in such systems.

Findings

Invariant tori persist under multiplicative noise with small perturbations.

Large deviation principles quantify the probability of trajectories deviating from invariant tori.

As noise intensity decreases, deviations become exponentially unlikely.

Abstract

This paper investigates the effect of random perturbations, in particular multiplicative noise, on the integrable structure of Hamiltonian systems, with a particular focus on KAM theory for stochastic Hamiltonian dynamics. We prove that, under suitable assumptions, for an integrable Hamiltonian system subject to both a small deterministic perturbation and multiplicative noise, the invariant tori with Diophantine frequencies persist in the sense of most probable paths. Furthermore, when the intensity of the multiplicative noise is sufficiently small, we use the large deviation principle to characterize the asymptotic probability of solution trajectories deviating from these invariant tori, and we derive the corresponding rate function.