A quenched local limit theorem for stochastic flows

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.

24 pages; fixed typos, added a reference, improved some exposition, and added new statement about temporal decorrelation of the spacetime-stationary solution in this version

References in corpus (5)

• An almost sure invariance principle for random walks in a space-time random environment
• Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation
• Quenched local limit theorem for random walks among time-dependent ergodic degenerate weights
• A forward-backward SDE from the 2D nonlinear stochastic heat equation
• Moderate deviations for diffusion in time dependent random media

Cited by in corpus (1)

• Random walk on nonnegative integers in beta distributed random environment