Title: 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 $\X$ 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 $\X$ for an asymptotically positive fraction of times $t$. If the ball grows as the logarithm of time, there are individual points in $\X$ whose images are eventually in the ball. Along the way, we prove several lemmas which may be of independent use. One is a slight generalization of the support theorem'', on the distributions of diffusion paths. Another, not heretofore unknown, but presented in a general and convenient form, is a bound the tails of the hitting times for one-dimensional stochastic processes with drift. If $X_n$ is a random walk with positive drift, and $\tau_x$ the first time that $X_n\geq x$, then it is well known that $\tau_x$ has subexponential tails. We show, under quite general conditions, that the same is true when the $\Ex X_{n+1}-X_n\cond \F_n\rb\geq c>0$.