  We consider the evolution of a connected set on the plane carried by a
periodic incompressible stochastic flow. While for almost every realization of
the random flow at time t most of the particles are at a distance of order
sqrt{t} away from the origin, there is a measure zero set of points, which
escape to infinity at the linear rate. We study the set of points visited by
the original set by time t, and show that such a set, when scaled down by the
factor of t, has a limiting non random shape.
