It has been suggested that stochastic .ows might be used to model
the spread 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 $x
\in \mathbb{R}^d$ 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.