A Critical Drift-Diffusion Equation: Connections to the Diffusion on $\textbf{SL}(2)$

TL;DR

This work builds a rigorous bridge between a planar drift-diffusion with a Gaussian divergence-free drift and a diffusion on the Lie group $SL(2)$. By constructing a canonical $SL(2)$ diffusion via $dF=F\,dB$ with an $SO(2)$-invariant covariance and coupling the environment $b$ to a Brownian driver $B$, the authors prove that the quenched drift $u$ is well approximated, in a scale-dependent sense, by $F^\dagger x$, after a logarithmic time-change $\tau(s)$. A two-scale homogenization with proxy correctors $\tilde{\phi}_L$ yields quantitative bounds comparing $u$ to $\tilde{u}_L$, and the coupling transfers intermittency from the non-Gaussian diffusion $F$ to $u$, producing anomalous moment growth and nontrivial regularity. The results illuminate a concrete link between stochastic homogenization in rough environments and diffusion processes on Lie groups, with implications for understanding large-scale transport in stochastic flows and random media.

Abstract

In this note, we connect two seemingly unrelated objects: On the one hand is a two-dimensional drift-diffusion process $X$ with divergence-free and time-independent drift $b$. The drift is given by a stationary Gaussian ensemble, and we focus on the critical case where a small-scale cut-off is necessary for well-posedness and the large-scale cancellations lead to a borderline super-diffusive behavior. On the other hand is the natural diffusion $F$ on the Lie group $\textbf{SL}(2)$ of matrices of determinant one. As a consequence of this connection, the strongly non-Gaussian character of $F$ transmits to how $X$ depends on its starting point.

Theorems & Definitions (11)

• Theorem 1
• Corollary 1
• Proposition 1
• Theorem 2: CMOW & MOW24
• Corollary 2
• Lemma 1
• Lemma 2
• proof : Proof of Theorem \ref{['thm:mr01']}.
• proof : Proof of Proposition \ref{['prop:mr02']}.
• Lemma 3: OW24