Speaker
Description
In applied mathematics, for instance through the work of Papanicolaou,
it has been known that convection may lead to a substantial increase of
the effective diffusivity, here of a passive tracer. We consider a diffusion process with a random time-independent and spatially stationary drift that de-correlates on large scales. The two-dimensional case is scaling-wise critical; we focus on a divergence-free drift, which can be written as the curl of what is known as the Gaussian free field. In the presence of a small-scale cut-off, we prove that the process is borderline super-diffusive: Its annealed second moments grow like $t\sqrt{\mathrm{ln}\,t}$ for $t \gg 1$. This refines older results of Tóth and Valkó and recent result of Cannizzaro, Haunschmid-Sibitz and Toninelli; the method however is completely different and appeals to quantitative stochastic homogenization of the generator that can be reformulated as a divergence-form second-order elliptic operator.
This is joint work with Chatzigeorgiou, Morfe, and Wang.
