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 $$\begin{eqnarray} \phi_0 (x) &=& x, \\ d\phi_t (x) &=& F (dt, \phi_t (x)),\end{eqnarray}$$ where $F(t, x)$ is a field of semimartingales on $x \in \mathbb{R}^d$ for $d \geq 2$ whose local characteristics are bounded and Lipschitz. The particles are points in a bounded set $\mathscr{X}$, and we ask how far the substance has spread in a time $T$. That is, we define $$\Phi^*_T = \sup_{x \in \mathscr{X}} \sup_{0 \leq t \leq T} || \phi_t (x) ||,$$ and seek to bound $P\{\Phi_T^*>z\}$.

Without drift, when $F(\cdot, x)$ are required to be martingales, although single points move on the order of $\sqrt{T}$, it is easy to construct examples in which the supremum $\Phi_T^*$ still grows linearly in time—that is, $\lim \inf_{T \to \infty}\Phi_T^* / T > 0$ almost surely. We show that this is an upper bound for the growth; that is, we compute a finite constant $K_0$, depending on the bounds for the local characteristics, such that $$ \limsup_{T \to \infty} \frac{\Phi^*_T}{T} \leq K_0 \text{ almost surely.}$$ A linear bound on growth holds even when the field itself includes a drift term.