This talk is about a simple but rich model problem at the cross section of stochastic homogenization and singular stochastic PDE: We consider a drift-diffusion process with a time-independent and divergence-free random drift that is of white-noise character. As already realized in the physics literature, the critical case of two space dimensions is most interesting: The elliptic generator requires a small-scale cut-off for well- posedness, and one expects marginally super-diffusive behavior on large scales. I will explain the criticality of the two-dimensional case in the introductory course by scaling arguments. […]

Mini-course: Flows of nonsmooth vector fields Consider a vector field v on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindelöf) Theorem states that, if the vector field is Lipschitz in space, for every initial datum x there is a unique trajectory γ starting at x at time 0 and solving the ODE γ'(t) = v(t, γ(t)). The theorem looses its validity as soon as v is slightly less regular. However, if we bundle all trajectories into a global map allowing x to vary, a celebrated theory started by DiPerna […]