Asymptotic support theorem for planar isotropic Brownian flows

Autores: Moritz Biskamp
Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 2, 2013, págs. 699-721
Idioma: inglés
DOI: 10.1214/11-AOP701
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Resumen
- It has been shown by various authors that the diameter of a given nontrivial bounded connected set 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 X⊆R2; 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.