In this work we use the stochastic flow decomposition technique to get components that represent the dynamics of the slow and fast motion of a stochastic differential equation with a random perturbation. Assuming a Lipschitz condition for vector fields and an average principle we get an approximation for the slow motion. To obtain the estimate for the rate of convergence we use a distance function which is defined in terms of the height functions associated to an isometric embedding of the manifold into the Euclidean space. This metric is topologically equivalent to the Riemannian distance given by the infimum of the lengths of all admissible curves between two points and works well with stochastic calculation tools. Finally, we get an estimate for the approximation between the solution of perturbed system and the original process provided by the unperturbed.

中文翻译：

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随机流的分解和慢扰动的平均原理

在这项工作中，我们使用随机流分解技术来获得代表具有随机扰动的随机微分方程的慢速和快速运动的动力学的分量。假设向量场的 Lipschitz 条件和平均原理，我们得到了慢动作的近似值。为了获得收敛速度的估计，我们使用距离函数，该函数根据与流形等距嵌入欧几里得空间相关联的高度函数定义。该度量在拓扑上等效于由两点之间所有可允许曲线的长度的下界给出的黎曼距离，并且适用于随机计算工具。最后，