Passive tracer in non-Markovian, Gaussian velocity field

We consider the trajectory of a tracer that is the solution of an ordinary differential equation $\dot{\text{bbX}}(t)=\text{bbV}(t,\text{bbX}(t)),X(0)=0$, 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 $\text{bbX}(t)\mathrm{/}\sqrt{t}$ converges in law, as $t\to +\mathrm{\infty }$, 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.