logt-Superdiffusivity for a Brownian particle in the curl of the 2D GFF

The present work is devoted to the study of large time behaviour a critical Brownian diffusion in two dimensions, whose drift divergence-free, ergodic and given by curl 2-dimensional Gaussian free field. We prove conjecture, made (J. Stat. Phys. 147 (2012) 113–131), according which coefficient D(t) diverges as logt for t→∞. Starting from fundamental Alder Wainwright (Phys. Rev. Lett. 18 (1967) 988–990), logarithmically superdiffusive has been predicted occur wide variety out-of-equilibrium systems spatial dimension d=2. Examples include tracer particle fluid, self-repelling polymers random walks, particles divergence-free environments and, more recently, Anisotropic KPZ equation. Even if all these cases it expected that D(t)∼logt, best authors’ knowledge, this first instance such precise asymptotics rigorously established.