A Critical Drift-Diffusion Equation: Intermittent Behavior via Geometric Brownian Motion on SL(n)

2025/11/19 by Morfe, Peter S., Otto, Felix, Wagner, Christian
Computer Science · Economics, Econometrics and Finance · #Advanced Mathematical Modeling in Engineering #Stochastic processes and financial applications #Nonlinear Dynamics and Pattern Formation

paper · doi:10.48550/arxiv.2511.15473

Abstract

This paper concerns the so-called diffusion in the curl of the 2d Gaussian free field, and its generalization to higher dimensions n ≥ 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 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.

Citations

• Superdiffusive Central Limit Theorem for the Stochastic Burgers Equation at the critical dimension
• A critical drift-diffusion equation: intermittent behavior
• Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift
• The Gaussian free-field as a stream function: continuum version of the scale-by-scale homogenization result
• An invariance principle for the 2d weakly self-repelling Brownian polymer
• The FBSDE approach to sine-Gordon up to 6π
• Anomalous dissipation and Euler flows
• Anomalous diffusion by fractal homogenization
• The Gaussian free-field as a stream function: asymptotics of effective diffusivity in infra-red cut-off
• Brownian particle in the curl of 2-d stochastic heat equations
• The Critical 2d Stochastic Heat Flow
• Large-scale regularity in stochastic homogenization with divergence-free drift
• The infinitesimal generator of the stochastic Burgers equation
• Superdiffusive bounds on self-repellent Brownian polymers and diffusion in the curl of the Gaussian free field in d=2

Related

• A critical drift-diffusion equation: intermittent behavior
• A Critical Drift-Diffusion Equation: Connections to the Diffusion on SL(2)
• The Gaussian free-field as a stream function: continuum version of the scale-by-scale homogenization result
• Brownian Motion with Drift on Spaces with Varying Dimension
• Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift