ON THE DISPERSION OF SETS UNDER THE ACTION OF AN ISOTROPIC BROWNIAN FLOW

We give a survey on results about the growth of the diameter of the image of a bounded subset of R^d under the action of a stochastic flow. We provide a new proof of the fact that, under reasonable assumptions, the diameter of this image set will almost surely grow at most linearly in time, and we establish an explicit upper bound for the linear growth rate which is both simpler and better than previous bounds. Our main tool is the Garsia–Rodemich–Rumsey Lemma.