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} $ 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 and the second moment of the displacement at a large time $t$ t behave like $2dc_{*}^{\nicefrac 12} t (\log t)^{\nicefrac 12}$ , in a quenched sense and with a precisely determined prefactor $c_{*}= c_{*}( \mathbb{P} )>0$ c ∗ = c ∗ ( P ) > 0 , independent of $\nu $ ν . 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.