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arXiv · 1007.1112

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 $φ$, 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. 2013-03-15. Asymptotic support theorem for planar isotropic Brownian flows. https://doi.org/10.1214/11-aop701

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