Diffusion with stochastic transport is investigated here when the random driving process is a very general Gaussian process including Fractional Brownian motion. The purpose is the comparison with a deterministic PDE which in certain cases represents the equation for the mean value. From this equation we observe a reduced dissipation property with respect to delta correlated noise when regularity is higher than the one of Brownian motion, a fact interpreted qualitatively here as a signature of the reduced dissipation observed for 2D turbulent fluids due to the inverse cascade. We give results also for the variance of the solution and for a scaling limit of a two-component noise input.