Asymptotic support theorem for planar isotropic Brownian flows

TL;DR

This paper proves an asymptotic support theorem for planar isotropic Brownian flows, showing that scaled trajectories converge to Lipschitz functions with a specific growth rate, extending previous diameter growth results.

Contribution

It extends diameter growth results to an asymptotic support theorem for trajectories of planar IBFs with positive Lyapunov exponent.

Findings

Trajectories converge in Hausdorff distance to Lipschitz functions.

The convergence is in probability.

The Lipschitz constant equals the known linear growth rate K.

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.