Title: Linear bounds for stochastic dispersion

Abstract: It has been suggested that stochastic flows might be used to model the spread of passive tracers in a turbulent fluid. We define a stochastic flow by the equations

0(x) = x ,

dt(x) = F(dt, t(x)),

where F(t,x) is a field of semimartingales on x R for d>1, whose local characteristics are bounded and Lipschitz. The particles are points in a bounded set X, and we ask how far the substance has spread in a time T. Without drift, when F( . ,x) are required to be martingales in other words, single points move on the order of T in time T. Nonetheless, it is easy to construct examples in which the supremum still grows linearly in time, almost surely. We show that this exhausts the possibilities for the growth: the growth can never be faster than linear, with a computable upper bound on the speed. A linear bound on growth holds even when the field itself includes a drift term.