On the Superdiffusive Behavior of Passive Tracer with a Gaussian Drift

In the present article we consider a motion of a passive tracer particle, whose trajectory satis es the It^o stochastic dierential equation dx(t) = V(t; x(t))dt + 2dw(t), where w() is a Brownian motion, V is a stationary Gaussian random eld with incompressible realizations independent of w() and > 0. We prove that when the drift fails to possess a stream matrix (in the sense of [19, 15]) the motion of the tracer particle is superdiusive. The result is shown both for steady (time independent) and time dependent Markovian elds. In addition, we provide explicit upper and lower bounds for the Hurst exponent. 1