A quenched local limit theorem for stochastic flows

TL;DR

This paper proves a quenched local limit theorem for particles in Gaussian random velocity fields, showing that their long-time density can be approximated by a deterministic Gaussian modulated by a stationary random field, with a special case for incompressible flows.

Contribution

It establishes a quenched local limit theorem for stochastic flows driven by Gaussian fields, extending classical results to random environments with specific conditions.

Findings

Quenched density approximated by Gaussian times a stationary field

Incompressible flows lead to a trivial stationary field U=1

Results hold in any Euclidean dimension

Abstract

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