Linear bounds for stochastic dispersion

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 Φ(x) = x, dΦt(x) = F(dt, Φt(x)), where F(t, x) is a field of semimartingales on x ∈ ℝd for d ≥ 2 whose local characteristics are bounded and Lipschitz. The particles are points in a bounded set script X sign, and we ask how far the substance has spread in a time T. That is, we define Φ*T = supx∈script T sign 0≤t≤Tsup ∥Φt(x)∥, and seek to bound P {Φ*T > z}. Without drift, when F(·, x) are required to be martingales, although single points move on the order of √T, it is easy to construct examples in which the supremum Φ*T still grows linearly in time - that is, lim inf T→∞ Φ*T/T > 0 almost surely. We show that this is an upper bound for the growth; that is, we compute a finite constant K0, depending on the bounds for the local characteristics, such that lim supT→∞ Φ*T/T ≤ K0 almost surely. A linear bound on growth holds even when the field itself includes a drift term.