Anomalous Diffusivity and Regularity for Random Incompressible Flows

Explore the long-time behavior of Brownian motion in stationary, incompressible random drift fields through this mathematical lecture by Scott Armstrong from Sorbonne Université. Examine how variance of displacement grows faster than linearly in time, with exponents determined by correlation structures as predicted by Bouchaud-Georges in 1990. Learn about the divergence-form drift-diffusion operator approach and discover how scale-by-scale coarse-graining schemes applied to coefficients produce effective Laplacians with scale-dependent diffusivity. Understand the rigorous mathematical framework that validates perturbative renormalization group heuristics originally proposed by Bouchaud-Georges. Investigate the crucial role of anomalous regularization and elliptic estimates that remain independent of bare molecular diffusivity. Gain insights into collaborative research conducted with A. Bou-Rabee and T. Kuusi, presented at the Institut des Hautes Etudes Scientifiques (IHES).