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

TL;DR

This paper explores the relationship between a critical two-dimensional drift-diffusion process with Gaussian random drift and the diffusion on the Lie group SL(2), revealing how non-Gaussian features influence the process's dependence on initial conditions.

Contribution

It establishes a novel connection between a critical drift-diffusion process and diffusion on SL(2), highlighting the impact of non-Gaussian characteristics on the process.

Findings

Critical drift-diffusion exhibits borderline super-diffusive behavior.

Non-Gaussian nature of SL(2) diffusion influences process dependence.

Small-scale cut-off is essential for well-posedness.

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.