具有互补分布流形中随机流的分解
Decomposition of stochastic flows in manifolds with complementary distributions
摘要：
设$M$是局部具有两个互补分布（水平分布和垂直分布）的微分流形.我们考虑在这些分布中的每一个中由向量场生成的$M$的两个子群（局部）同构。给定$M$的一个随机流$φ_t$，在初始条件的一个邻域内，直到一个停时，我们分解了$φ_t = ξ_t \circ ψ_t$，其中第一个分量是水平自同态群中的一个扩散，第二个分量是垂直自同态群中的一个过程.进一步的分解将包括两个以上的组件;它导致局部坐标中的最大级联分解，其中每个组件仅在相应的坐标中起作用。
英文摘要：
Let $M$ be a differentiable manifold endowed locally with two complementary distributions, say horizontal and vertical. We consider the two subgroups of (local) diffeomorphisms of $M$ generated by vector fields in each of of these distributions. Given a stochastic flow $φ_t$ of diffeomorphisms of $M$, in a neighbourhood of initial condition, up to a stopping time we decompose $φ_t = ξ_t \circ ψ_t$ where the first component is a diffusion in the group of horizontal diffeomorphisms and the second component is a process in the group of vertical diffeomorphisms. Further decomposition will include more than two components; it leads to a maximal cascade decomposition in local coordinates where each component acts only in the corresponding coordinate.
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