Uniform shrinking and expansion under isotropic Brownian flows

TL;DR

This paper investigates the finite-time transport properties of isotropic Brownian flows, demonstrating conditions under which uniform shrinking or expansion occurs, and analyzing the long-term behavior of curves within these flows.

Contribution

It establishes positive probability results for uniform shrinking or expansion of balls and provides a control theorem for flows with drift, advancing understanding of isotropic Brownian flow dynamics.

Findings

Uniform shrinking or expansion of balls can occur with positive probability

The length of a curve with negative Lyapunov exponent tends to zero over time

Control theorem for isotropic Brownian flows with drift

Abstract

We study some finite time transport properties of isotropic Brownian flows. Under a certain nondegeneracy condition on the potential spectral measure, we prove that uniform shrinking or expansion of balls under the flow over some bounded time interval can happen with positive probability. We also provide a control theorem for isotropic Brownian flows with drift. Finally, we apply the above results to show that under the nondegeneracy condition the length of a rectifiable curve evolving in an isotropic Brownian flow with strictly negative top Lyapunov exponent converges to zero as $t\to \infty$ with positive probability.