## Linear bounds for stochastic dispersion.(English)Zbl 1044.60055

Summary: 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 $\varphi_0(x)=x,\quad d\varphi_t(x)=F \bigl(dt,\varphi_t (x) \bigr),$ 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 $${\mathcal X}$$, and we ask how far the substance has spread in a time $$T$$. That is, we define $\Phi^*_T=\sup_{x\in{\mathcal X}}\sup_{0 \leq t\leq T}\bigl\| \varphi_t (x)\bigr\|,$ 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 $$\limsup_{T\to\infty} (\Phi^*_T/T)\leq K_0\text{ almost surely}.$$ A linear bound on growth holds even when the field itself includes a drift term.

### MSC:

 60H30 Applications of stochastic analysis (to PDEs, etc.) 60H20 Stochastic integral equations 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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