Diffusion in the curl of the two-dimensional Gaussian Free Field
I will discuss the large time behaviour of a Brownian diffusion in two dimensions, whose drift is divergence-free and ergodic, and given by the curl of the two-dimensional Gaussian Free Field.
At first, we prove the conjecture by B. Tóth and B. Valkó that the mean square displacement is of order $t \sqrt{\log t}$. The same type of superdiffusive behaviour has been predicted to occur for a wide variety of (self)-interacting diffusions in dimension d = 2, including the diffusion of a tracer particle in a fluid, self-repelling polymers and random walks, Brownian particles in divergence-free random environments, and, more recently, the 2-dimensional critical Anisotropic KPZ equation. Then, we show that, by suitably taming the strength of the drift (in the so-called weak coupling scaling), it is possible to derive a homogenization type result and show that a Donsker's invariance principle indeed holds.
This is based joint work(s) with H. Giles, L. Haundschmid and F. Toninelli.