Uniform Shrinking and Expansion under Isotropic Brownian Flows

Peter Baxendale and Georgi Dimitroff ()
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Peter Baxendale: University of Southern California
Georgi Dimitroff: Fraunhofer ITWM

Journal of Theoretical Probability, 2009, vol. 22, issue 3, 620-639

Abstract: 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.

Keywords: Stochastic differential equation; Stochastic flow of diffeomorphisms; Isotropic Brownian flow; Cameron–Martin space; Reproducing kernel; Control theorem; 37H10; 60H10; 46E22; 60G15; 60G60 (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1007/s10959-008-0193-3

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