In this paper, we generalize the classical Freidlin-Wentzell's theorem for random perturbations of Hamiltonian systems. In stead of the two-dimensional standard Brownian motion, the coefficient for the noise term is no longer the identity matrix but a state-dependent matrix plus a state-dependent matrix that converges uniformly to 0 on any compact sets as $\epsilon$ tends to 0. We also take the drift term into consideration where the drfit term also contains two parts, the state-dependent mapping and a state-dependent mapping that converges uniformly to 0 on any compact sets as $\epsilon$ tends to 0. In the proof, we use the result of generalized differential operator. We also adapt a new way to prove the weak convergence inside the edge by constructing an auxiliary process and apply Girsanov's theorem in the proof of gluing condition.

中文翻译：

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关于哈密顿系统平均的 Freidlin-Wentcell 定理的推广

在本文中，我们将经典的 Freidlin-Wentzell 定理推广到哈密顿系统的随机扰动。代替二维标准布朗运动，噪声项的系数不再是单位矩阵，而是一个状态相关矩阵加上一个状态相关矩阵，该矩阵在任何紧集上一致收敛到 0，因为 $\epsilon$ 趋于到 0。我们还考虑了漂移项，其中 drfit 项也包含两部分，状态相关映射和状态相关映射，当 $\epsilon$ 趋于 0 时，它在任何紧凑集上一致收敛到 0。证明，我们使用广义微分算子的结果。我们还采用了一种新的方法来证明边缘内部的弱收敛性，通过构造一个辅助过程并将 Girsanov 定理应用于粘合条件的证明。