On positivity of the variance of a tracer moving in a divergence-free Gaussian random field

We consider the trajectory of a tracer that is the solution of an ordinary differential equation Ẋ(t)=V(t,X(t)), with the right hand side, that is a stationary, zero-mean, Gaussian vector field with incompressible realizations. It is known, see Komorowski and Papanicolaou (1997), that X(t)/t converges in law, as t→+∞, to a normal vector N(0,κ), provided that the covariance matrix of the field is compactly supported in t. The question whether the limiting diffusivity matrix vanishes or not has been left open. In the present note we formulate a sufficient condition for the matrix κ to be non-vanishing.