Onsager-Machlup Functional and Large Deviation Principle for Stochastic Hamiltonian Systems

Abstract: This paper investigates the application of KAM theory within the probabilistic framework of stochastic Hamiltonian systems. We begin by deriving the Onsager-Machlup functional for the stochastic Hamiltonian system, identifying the most probable transition path for system trajectories. By establishing a large deviation principle for the system, we derive a rate function that quantifies the system's deviation from the most probable path, particularly in the context of rare events. Furthermore, leveraging classical KAM theory, we demonstrate that invariant tori remain stable, in the most probable sense, despite small random perturbations. Notably, we establish that the exponential decay rate for deviations from these persistent tori coincides precisely with the previously derived rate function, providing a quantitative probabilistic characterization of quasi-periodic motions in stochastic settings.