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.
In the presence of an (artificial) large-scale cut-off at scale L, as a consequence of standard stochastic homogenization theory and its notion of a corrector, there exist harmonic coordinates with a stationary gradient F ; the merit of these coordinates being that under their lens, the drift-diffusion process turns into a martingale. I will revisit (qualitative) stochastic homogenization theory
in the introductory course.
It has recently been established that the second moments diverge as 𝔼 |F|² ~ √ln L as L ↑ ∞. We show that in this limit, |F|²/𝔼 |F|² is not equi-integrable, while |det F |/𝔼 |F|² converges to zero (in probability). This suggests that any limit, if it exists, will not admit a simple characterization.
We establish this asymptotic behavior by characterizing a proxy F̃ introduced in previous work as the solution of an Itô SDE w. r. t. the variable L, and which implements the concept of a scale-by-scale homogenization. For this proxy, we establish 𝔼|F̃ |⁴ ≫ (𝔼|F̃ |²)² and 𝔼(det F̃ )² ≲ 1. In view of the former property, we assimilate this phenomenon to intermittency. I will try to elucidate the idea of scale-by-scale homogenization in the introductory course.
This is joint work with G. Chatzigeorgiou, P. Morfe, L. Wang, and with C. Wagner.