Mathematics – Probability
Scientific paper
2010-07-07
Mathematics
Probability
Scientific paper
10.1214/11-AOP701
It has been shown by various authors that the diameter of a given non-trivial bounded connected set $\mathcal{X}$ grows linearly in time under the action of an isotropic Brownian flow (IBF), which has a non-negative 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 non-trivial compact connected set $\mathcal{X} \subseteq \R^2$, i.e. 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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