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 ẋ(t)=F(x(t)), x(0)=x, 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.