Dispersion of volume under the action of isotropic Brownian flows

TL;DR

This paper investigates the transport properties of isotropic Brownian flows, demonstrating asymptotic normality of transported measures and volumes, and showing that the volume of images of open sets converges to a positive random variable.

Contribution

It establishes conditions for asymptotic normality of measures and volumes under isotropic Brownian flows, and proves convergence of volume martingales to positive limits.

Findings

Asymptotic normality of the image of finite measures.

Asymptotic normality of the volume of images of sets.

Almost sure positivity of volume limits for certain flows.

Abstract

We study transport properties of isotropic Brownian flows. Under a transience condition for the two-point motion, we show asymptotic normality of the image of a finite measure under the flow and -- under slightly stronger assumptions -- asymptotic normality of the distribution of the volume of the image of a set under the flow. Finally, we show that for a class of isotropic flows, the volume of the image of a nonempty open set (which is a martingale) converges to a random variable which is almost surely strictly positive.