| Summary: | 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 χ 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 A. The linear growth may further be identified (again, almost surely), with a much stronger behavior, which we call "ball-chasing": if ψ is any path with Lipschitz constant smaller than A, the ball of radius e around ψ (t) contains points of the image of χ for an asymptotically positive fraction of times t. If the ball grows as the logarithm of time, there are individual points in χ whose images eventually remain in the ball.
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