The Annals of Probability

Asymptotic support theorem for planar isotropic Brownian flows

Moritz Biskamp

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Abstract

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 $K$.

Article information

Source
Ann. Probab., Volume 41, Number 2 (2013), 699-721.

Dates
First available in Project Euclid: 8 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1362750939

Digital Object Identifier
doi:10.1214/11-AOP701

Mathematical Reviews number (MathSciNet)
MR3077523

Zentralblatt MATH identifier
1277.60069

Subjects
Primary: 60G17: Sample path properties 37C10: Vector fields, flows, ordinary differential equations
Secondary: 60G15: Gaussian processes 37H10: Generation, random and stochastic difference and differential equations [See also 34F05, 34K50, 60H10, 60H15]

Keywords
Stochastic flows isotropic Brownian flows asymptotic expansion asymptotic support theorem

Citation

Biskamp, Moritz. Asymptotic support theorem for planar isotropic Brownian flows. Ann. Probab. 41 (2013), no. 2, 699--721. doi:10.1214/11-AOP701. https://projecteuclid.org/euclid.aop/1362750939


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