Most Probable KAM Tori in Stochastic Hamiltonian Systems

This paper investigates in depth how stochastic perturbations affect the integrable structure of Hamiltonian systems and develops a KAM theory for stochastic Hamiltonian dynamics, in the sense of the most probable path. We first derive the Onsager-Machlup functional for stochastic Hamiltonian systems driven by time-dependent noise coefficients and identify the most probable path of the system trajectories. Building on this, we establish a large deviation principle and obtain an explicit rate function that quantitatively characterizes trajectory deviations, in particular for rare events. The main contribution of this work is to prove that, under stochastic noise, the original quasi-periodic invariant tori persist in the sense of the most probable path, thereby demonstrating the stability of KAM structures in random environments. Moreover, we show that the Onsager-Machlup functional coincides exactly with the large deviation rate function, thereby providing a quantitative characterization of both the structural persistence of quasi-periodic motions and the geometry of fluctuations in stochastic Hamiltonian systems. Overall, our results extend the classical KAM framework to stochastic settings and offer new insight into the behavior of complex dynamical systems under noise.