Uniform Shrinking and Expansion under Isotropic Brownian Flows |
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Authors: | Peter Baxendale Georgi Dimitroff |
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Institution: | (1) Department of Mathematics, University of Southern California, 3620 S. Vermont Avenue, Los Angeles, CA 90089-2532, USA;(2) Fraunhofer ITWM, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany |
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Abstract: | We study some finite time transport properties of isotropic Brownian flows. Under a certain nondegeneracy condition on the
potential spectral measure, we prove that uniform shrinking or expansion of balls under the flow over some bounded time interval
can happen with positive probability. We also provide a control theorem for isotropic Brownian flows with drift. Finally,
we apply the above results to show that, under the nondegeneracy condition, the length of a rectifiable curve evolving in
an isotropic Brownian flow with strictly negative top Lyapunov exponent converges to zero as t→∞ with positive probability.
P. Baxendale’s research was supported in part by NSF Grant DMS-05-04853. |
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Keywords: | Stochastic differential equation Stochastic flow of diffeomorphisms Isotropic Brownian flow Cameron– Martin space Reproducing kernel Control theorem |
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