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

TL;DR

This paper explores the diffusion process related to the 2D Gaussian free field and its higher-dimensional generalizations, revealing a connection with geometric Brownian motion on SL(n) that explains intermittent behaviors.

Contribution

It introduces a reformulation of the homogenization scheme using SDEs, uncovering a novel link with geometric Brownian motion on SL(n) that advances understanding of intermittency.

Findings

Connection between diffusion in Gaussian free fields and geometric Brownian motion on SL(n)

Insight into intermittent behavior of the Lagrangian coordinate

Reformulation of homogenization scheme in terms of SDEs

Abstract

This paper concerns the so-called diffusion in the curl of the 2d Gaussian free field, and its generalization to higher dimensions $n \geq 2$, 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 $\textbf{SL}(n)$. The analysis of this process sheds light on the original problem, particularly as it pertains to intermittent behavior exhibited by the (averaged) Lagrangian coordinate.