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March 2013 Asymptotic support theorem for planar isotropic Brownian flows
Moritz Biskamp
Ann. Probab. 41(2): 699-721 (March 2013). DOI: 10.1214/11-AOP701


It has been shown by various authors that the diameter of a given nontrivial bounded connected set $\mathcal{X}$ grows linearly in time under the action of an isotropic Brownian flow (IBF), which has a nonnegative top-Lyapunov exponent. In case of a planar IBF with a positive top-Lyapunov exponent, the precise deterministic linear growth rate $K$ of the diameter is known to exist. In this paper we will extend this result to an asymptotic support theorem for the time-scaled trajectories of a planar IBF $\varphi$, which has a positive top-Lyapunov exponent, starting in a nontrivial compact connected set $\mathcal{X}\subseteq\mathbf{R}^{2}$; that is, we will show convergence in probability of the set of time-scaled trajectories in the Hausdorff distance to the set of Lipschitz continuous functions on $[0,1]$ starting in $0$ with Lipschitz constant $K$.


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Moritz Biskamp. "Asymptotic support theorem for planar isotropic Brownian flows." Ann. Probab. 41 (2) 699 - 721, March 2013.


Published: March 2013
First available in Project Euclid: 8 March 2013

zbMATH: 1277.60069
MathSciNet: MR3077523
Digital Object Identifier: 10.1214/11-AOP701

Primary: 37C10 , 60G17
Secondary: 37H10 , 60G15

Keywords: asymptotic expansion , asymptotic support theorem , isotropic Brownian flows , Stochastic flows

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 2 • March 2013
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