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

Abstract

This paper investigates the effect of state-dependent multiplicative noise on the integrable structure of Hamiltonian systems. Under a local high-dimensional Lamperti-type condition, we derive the Onsager--Machlup functional in the corresponding flattening coordinates. When the diffusion vector fields are Hamiltonian, the resulting action is minimized uniquely by the deterministic Hamiltonian trajectory. Combining this characterization with a finitely differentiable KAM theorem, we prove that, under suitable assumptions, the invariant tori with Diophantine frequencies of an integrable Hamiltonian system persist, in the sense of most probable paths, under a small deterministic perturbation and multiplicative noise. Furthermore, as the intensity of the multiplicative noise tends to zero, we establish a localized large deviation principle that characterizes the asymptotic probability of solution trajectories deviating from these invariant tori.