A Critical Drift-Diffusion Equation: Intermittent Behavior via Geometric Brownian Motion on $\text{SL}(n)$

Abstract

This paper concerns the so-called diffusion in the curl of the 2d Gaussian free field, and its generalization to higher dimensions $n\ge 2$, building on the scale-by-scale homogenization approach developed recently by Chatzigeorgiou, Morfe, Otto, and Wang [13]. It begins by reformulating the approximation scheme of that work in terms of SDEs in the length scale $L$. This exposes an unexpected connection with a certain geometric Brownian motion on the special linear group $\text{SL}(n)$. The analysis of this process sheds light on the original problem, particularly as it pertains to intermittent behavior exhibited by the (averaged) Lagrangian coordinate.

This preprint subsumes and significantly improves our previous arXiv posts arXiv:2404.00709, arXiv:2404.13641, and arXiv:2410.15983. Only this one will be submitted for publication in a journal