This paper proposes an effective approximation result for the behavior of a small transversal perturbation to a completely integrable stochastic Hamiltonian system on a symplectic manifold. We derive an averaged stochastic differential equations (SDEs) in the action space for the action component of the perturbed system, where the averaged drift coefficient is characterized by the averages of that of the action component with respect to the invariant measure of the unperturbed system on the corresponding invariant manifolds. Then, the averaging principle is shown to be valid such that the action component of the perturbed system converges to the solution of averaged SDEs in the mean square sense.

本文提出了辛流形上完全可积随机哈密顿系统的小横向扰动行为的有效近似结果。我们在作用空间中推导出扰动系统的作用分量的平均随机微分方程（SDE），其中平均漂移系数由作用分量相对于未扰动系统的不变测度的平均值来表征相应的不变流形。然后，平均原理被证明是有效的，使得扰动系统的作用分量收敛到均方意义上的平均 SDE 的解。