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Explore the mathematical foundations of anomalous diffusion in this comprehensive lecture from the Centre International de Rencontres Mathématiques. Delve into the fundamental ansatz of phenomenological theories of passive scalar turbulence as the speaker constructs a class of incompressible vector fields that exhibit properties observed in fully turbulent velocity fields. Learn how the associated scalar advection-diffusion equation generically displays anomalous diffusion behavior and discover an analytical framework for studying this phenomenon through a backward cascade of renormalized eddy viscosities. Examine the proof methodology based on "fractal" homogenization, which involves performing a cascade of homogenizations across arbitrarily many length scales. This lecture was recorded during the thematic meeting "Physics and Mathematics of hydrodynamic and wave turbulence" in May 2025 at CIRM in Marseille, France, providing insights into cutting-edge research at the intersection of mathematical analysis and fluid dynamics.
