Passive scalar motion in a family of random Gaussian velocity fields with long-range correlations is shown to converge to persistent fractional Brownian motions in long times.

We establish diffusion and fractional Brownian motion approximations for motions in a Markovian Gaussian random field with a nonzero mean.

We prove the Taylor-Kubo formula for a class of isotropic, non-mixing flows with long-range correlation. For the proof, we develop the method of high order correctors expansion.