📝 SummarXiv

Abstract

This paper conducts an in-depth investigation into the impact of stochastic perturbations-particularly multiplicative noise-on the integrable structures of Hamiltonian systems, with a central focus on developing a KAM theory for stochastic Hamiltonian dynamics. We begin by deriving the Onsager-Machlup functional for Hamiltonian systems driven by multiplicative noise and identifying the most probable transition paths of system trajectories. This analysis reveals the fundamental differences in how additive versus multiplicative noise influences the integrability of Hamiltonian systems. Building upon this, we establish a large deviation principle for the system and derive a rate function that quantitatively characterizes trajectory deviations, especially in the regime of rare events. The main contribution of this work lies in demonstrating that, under the small noise limit, the quasi-periodic invariant tori of the unperturbed system persist in a probabilistic sense, indicating the stability of KAM structures under stochastic perturbations. Furthermore, we show that the exponential rate of deviation from the invariant tori exactly matches the large deviation rate function, thus providing a quantitative characterization of the structural persistence and fluctuation geometry of quasi-periodic motions in stochastic Hamiltonian systems. These results extend the classical KAM framework to a stochastic setting and offer new insights into the behavior of complex dynamical systems under the influence of noise.