流形上随机流的渐近平坦性(英文)

Asymptotic Flatness of Stochastic Flow on Manifolds

中文关键词:  扩散过程  随机流  Hausdorff测度  流形

英文关键词:diffusion process  stochastic flow  Hausdorff measure  manifold

中文摘要:

      本文的目的是讨论流形上由随机微分方程确定的扩散过程的体积零化性质。令X_t(x)是描述流形M上的微分同胚流x→X_t(x)的扩散过程,K是M中具有正有限Hausdorff测度的紧致曲面,我们给出X_t(K)的面积在t→∞时几乎必然趋于零的条件,特别地,随机流X_t(·)的渐近零化定向可求长弧r:[0,1]→M的弧长。

英文摘要:

      The aim of this article is to discuss a volume nullification property of the diffusion process determined by a stochastic differential equation on a manifold. Let X_t(x) be a diffusion process describing a flow of diffeomorphisms x→X_t (x) in a manifold M, and K be a compact surface in M with positive finite Hausdorff measure. We present conditions under which the area of X_t(K) goes to zero almost surely and in moments as t→∞, in particular, the flow X_t(·) asymptotic nullifies the arc-lenth of oriented rectifiable arcs r:[0,1]→M.