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

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.