The Gaussian free field as a stream function: asymptotics of effective diffusivity in infra-red cut-off

We investigate the effective diffusivity of a random drift-diffusion operator that is at the borderline of standard stochastic homogenization theory: In two space-dimensions, we consider the divergence-free drift with stream function given by the Gaussian free-field, with an ultra-violet cut-off at scale unity and an infra-red cut-off at a scale L. We establish the precise scaling of how the effective diffusivity diverges in terms of L, specifying recent results based on a Wiener chaos decomposition and a mathematical physics-type analysis in the corresponding Fock space. This amounts to the study of convection-enhanced diffusion at the borderline to anomalous diffusion. It provides a quantitative stochastic homogenization perspective, and therefore yields quenched rather than annealed results. Joint work with Georgiana Chatzigeorgiou, Peter Morfe and Felix Otto (MPI).