  We consider the evolution of a connected set in Euclidean space 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 [DKK1], there is an uncountable set of measure
zero of points, which escape to infinity at the linear rate [CSS1]. In this
paper we prove that this set of linear escape points has full Hausdorff
dimension.
