Chasing balls through martingale fields

Abstract

We consider the way sets are dispersed by the action of stochastic flows derived from martingale fields. Under fairly general continuity and ellipticity conditions, the following dichotomy result is shown: any nontrivial connected set $\mathcal{X}$ either contracts to a point under the action of the flow, or its diameter grows linearly in time, with speed at least a positive deterministic constant $\gL$. The linear growth may further be identified (again, almost surely), with a much stronger behavior, which we call "ball-chasing'': if $\psi$ is any path with Lipschitz constant smaller than $\gL$, the ball of radius $\gep$ around $\psi(t)$ contains points of the image of $\mathcal{X}$ for an asymptotically positive fraction of times $t$. If the ball grows as the logarithm of time, there are individual points in $\mathcal{X}$ whose images eventually remain in the ball.