Uniform Shrinking and Expansion under Isotropic Brownian Flows

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 pr...

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Bibliographic Details
Published in:Journal of theoretical probability 2009-09, Vol.22 (3), p.620-639
Main Authors: Baxendale, Peter, Dimitroff, Georgi
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
Probability Theory and Stochastic Processes
Statistics
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Summary: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.
ISSN:0894-9840
1572-9230
DOI:10.1007/s10959-008-0193-3