| Asymptotic Flatness of Stochastic Flow on Manifolds |
| Received:June 03, 2002 |
| Key Words:
diffusion process stochastic flow Hausdorff measure manifold
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| Fund Project:Supported by the kcy Projcct of Chinese Minstry of Education and Supported by Natural Science Foundation of Beijing (1022004) |
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| Abstract: |
| 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 diffeomorphisms 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 r:[0,1]→M. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2004.02.001 |
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