Linear bounds for stochastic dispersion

It has been suggested that stochastic flows might be used to model the spre ad of passive tracers in a turbulent fluid. We define a stochastic flow by the equations phi (0)(x) = x, d phi (t)(x) = F(dt, phi (t)(x)), where F(t, x) is a field of semimartingales on x is an element of R-d for d greater than or equal to 2 whose local characteristics are bounded and Lip schitz. The particles are points in a bounded set X, and we ask how far the substance has spread in a time T. That is, we define Phi (T)* = sup(x is an element ofX) sup(0 less than or equal tot less than or equal toT) //phi (t)(x)// and seek to bound P{Phi (T)* > z). Without drift, when F(., x) are required to be martingales, although single points move on the order of rootT, it is easy to construct examples in whi ch the supremum Phi (T)* Still grows linearly in time-that is,lim inf T --> infinity Phi (T)*/T > 0 almost surely. We show that this is an upper bound for the growth; that is, we compute a finite constant K-0, depending on th e bounds for the local characteristics, such that lim sup(T --> infinity) Phi (T)*/T less than or equal to K-0 almost surely. A linear bound on growth holds even when the field itself includes a drift term.