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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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