Linear and near - Linear Bounds for Stochasticdispersionmike
نویسنده
چکیده
It has been suggested that stochastic ows might be used to model the spread of a passive substance on the surface of a body of water. We deene a stochastic ow by the equations 0 (x) = x; ddt(x) = F (dt; t(x)); where F (t; x) is a eld of semimartingales on x 2 R d for d 2 whose local characteristics are bounded and Lipschitz. The particles are points in a bounded set X, and we ask how far the substance has spread in a time T. That is, we deene T = sup x2X sup 0tT t(x) ; and seek to bound P T > z. If the eld F has a directed drift, of course, kt(x)k will grow linearly with time. Taking the maximum does not increase the growth rate very much: we show that for any > 1, limsup T!1 T T (log T) = 0 a.s. Without drift, when F (; x) are required to be martingales, although single points move on the order of p T , it is easy to construct examples in which the supremum T still grows linearly in time | that is, liminf T!1 T =T > 0 almost surely. We show that this is an upper bound for the growth; that is, we compute a nite constant K 0 , depending on the local characteristics, such that limsup T!1 T T K 0 a.s.
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تاریخ انتشار 1998