On superdiffusive behavior of a passive tracer in a random flow

In this note we consider a passive tracer model describing particle dispersion in a turbulent flow. The trajectory of the particle is given by the solution of an ordinary differential equation x ̇ ( t ) = F ( x ( t ) ) , x ( 0 ) = x 0 , where F ( x ) is a divergence-free, random vector field that is spatially homogeneous and isotropic. We show that trajectories of the tracer display superdiffusive behavior when the random velocity F ( x ) decorrelates, at large distances, but does it not rapidly but rather at some moderate rate. The main tools used in the proofs are variational principles and Tauberian-type theorems.