Passive tracer in non-Markovian, Gaussian velocity field

Abstract

We consider the trajectory of a tracer that is the solution of an ordinary differential equation $\dot\bbX(t)=\bbV(t, \bbX(t)),\ X(0)=0$, with the right hand side, that is a stationary, zero-mean, Gaussian vector field with incompressible realizations. It is known, see [K-F;C-X;K-L-O], that $\bbX(t)/\sqrt{t}$ converges in law, as $t\to+\infty$, to a normal, zero mean vector, provided that the field $V(t,x)$ is Markovian and has the spectral gap property. We wish to extend this result to the case when the field is not Markovian and its covariance matrix is given by a completely monotone Bernstein function.