CAJ | 학술논문

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Xt(x)是描述流形M上的微分同胚流x→Xt(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Xt(K)的面积在t→∞时几乎必然趋于零的条件,特别地,随机流Xt(·)的渐近零化定向可求长弧γ:[0,1]→M的弧长.
본문적목적시토론류형상유수궤미분방정학정적확산과정적체적령화성질.령Xt(x)시묘술류형M상적미분동배류x→Xt(x)적확산과정,K시M중구유정유한Hausdorff측도적긴치곡면.아문급출Xt(K)적면적재t→∞시궤호필연추우령적조건,특별지,수궤류Xt(·)적점근령화정향가구장호γ:[0,1]→M적호장.
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 Xt (x) be a diffusion process describing a flow of diffcomorphisms x→Xt (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 Xt(K) goes to zero almost surely and in moments as t →∞, in particular, the flow Xt(.) asymptotic nullifies the arc-lenth of oriented rectifiable arcs γ: [0, 1] → M.