A quenched local limit theorem for stochastic flows

We consider a particle undergoing Brownian motion in Euclidean space of any dimension, forced by Gaussian random velocity field that is white time and smooth space. show conditional on the field, quenched density after long can be approximated pointwise product deterministic spacetime-stationary $U$. If additionally assumed to incompressible, then $U\equiv 1$ almost surely we obtain local central limit theorem.