Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift

Abstract

We consider the long-time behavior of a diffusion process on ${\mathbb{R}}^d$ advected by a stationary random vector field which is assumed to be divergence-free, dihedrally symmetric in law and have a log-correlated potential. A special case includes ${\nabla }^{\perp }$ of the Gaussian free field in two dimensions. We show the variance of the diffusion process at a large time $t$ behaves like $2c_{\ast }t(\log t{)}^{1/2}$, in a quenched sense and with a precisely determined, universal prefactor constant $c_{\ast }>0$. We also prove a quenched invariance principle under this superdiffusive scaling. The proof is based on a rigorous renormalization group argument in which we inductively analyze coarse-grained diffusivities, scale-by-scale. Our analysis leads to sharp homogenization and large-scale regularity estimates on the infinitesimal generator, which are subsequently transferred into quantitative information on the process.

164 pages, 2 figures; previously announced at https://www.scottnarmstrong.com/2024/04/superdiffusive-CLT/