A Critical Drift-Diffusion Equation: Intermittent Behavior via Geometric Brownian Motion on $ \textbf{SL}(n)$
Peter S. Morfe, Felix Otto, Christian Wagner
TL;DR
This work identifies a scale-by-scale homogenization mechanism for a diffusion in a scale-critical, divergence-free Gaussian drift, revealing that the averaged Lagrangian coordinate $u$ can be approximated by a geometric Brownian flow on the Lie group $\textbf{SL}(n)$. By constructing an infra-red cut-off and a two-scale expansion, the authors relate the gradient dynamics $\nabla u$ to a Brownian driver on $\mathfrak{sl}(n)$, producing a Stratonovich SDE for a flow $F$ on $\textbf{SL}(n)$ with left-invariant dynamics. In 2D, this framework yields precise intermittency results: moments of $F$ grow superdiffusively, the top Lyapunov exponent equals $\tfrac{1}{4}$, and the Bessel-type process $R$ exhibits non-Gaussian tails due to rare, long-level-line excursions of the underlying Gaussian free field. The combination of scale-by-scale homogenization, Brownian motion on Lie algebras, and Lyapunov analysis provides a rigorous, quantitative bridge between stochastic homogenization and geometric stochastic flows, highlighting the geometric origin of intermittency in a critical scaling regime.
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.