Long Time Behaviour of Periodic Stochastic Flows

We consider the evolution of a set carried by a space periodic incompressible stochastic flow in a Euclidean space. We report on three main results obtained in [8, 9, 10] concerning long time behaviour for a typical realization of the stochastic flow. First, at time t most of the particles are at a distance of order √ t away from the origin. Moreover, we prove a Central Limit Theorem for the evolution of a measure carried by the flow, which holds for almost every realization of the flow. Second, we show the existence of a zero measure full Hausdorff dimension set of points, which escape to infinity at a linear rate. Third, in the 2-dimensional case, we study the set of points visited by the original set by time t. Such a set, when scaled down by the factor of t, has a limiting non random shape.