14/05/2025, 17:00 — 18:00 — Online
Scott Armstrong, CNRS Directeur de recherche at Sorbonne University and Courant Institute of Mathematical Sciences at NYU
Superdiffusivity for a diffusion in a critically-correlated incompressible random drift
We consider a Brownian particle in a divergence-free drift, where the vector field is a stationary random field exhibiting "critical" correlations. Predictions from physicists in the 80s state that, almost surely, this process should behave like a "sped-up" Brownian motion at large scales, with variance at time~$t$ being of order~$t \sqrt{\log t}$. In joint work with Ahmed Bou-Rabee and Tuomo Kuusi, we give a rigorous proof of this prediction using an iterative quantitative homogenization procedure, which is a way of formalizing a renormalization group argument. We consider the generator of the process and coarse-grain this operator, scale-by-scale, across an infinite number of scales. The random swirls of the vector field at each scale enhance the effective diffusivity. As we zoom out, we obtain an ODE for the effective diffusivity as a function of the scale, to deduce that it diverges at the predicted rate. Meanwhile, new coarse-graining arguments allow us to rigorously (and quenchedly) integrate out the smaller scales and prove the scaling limit.
