Anomalous scaling limit of a Brownian particle in a log-correlated potential

Scott Armstrong, Ahmed Bou-Rabee, Tuomo Kuusi

math.PR Oct 8, 2026 · v1 math-ph math.AP

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TL;DR

Main Theorems A–C and supporting results were formalized and machine-checked in Lean 4, with a paper-to-Lean correspondence in a (private) repository.

Abstract

We study a Brownian particle in $\mathbb{R}^d$, $d\geq2$, with drift given by the gradient of a log-correlated Gaussian potential. At weak disorder, we prove convergence to a scaling limit whose law is singular with respect to Brownian motion and whose invariant measure is given by Gaussian multiplicative chaos. The proof uses a renormalization group induction: at each scale, the elliptic operator is well approximated by a Laplacian with effective diffusivity decaying as a power of scale. We compute this exponent exactly in dimension two.

Problem

The paper studies the long-time behavior of a Brownian particle in R^d, d≥2, whose drift is the gradient of a log-correlated Gaussian potential. Effective diffusivity vanishes on large scales, so the particle is subdiffusive.

Approach

A renormalization group induction over scales uses coarse-grained matrices from Armstrong–Kuusi. At each scale the elliptic operator is approximated by a Laplacian with effective diffusivity decaying as a power of scale. Large-scale regularity, Dirichlet homogenization and a uniqueness argument for limiting Dirichlet forms are used to construct the scaling limit. Theorems A–C and their supporting results were formalized and machine-checked in Lean 4.

Results

At weak disorder, the rescaled process converges to a scaling limit whose law is singular with respect to Brownian motion, with Gaussian multiplicative chaos as invariant measure. In dimension two the diffusivity exponent is computed exactly via the identity ā_m = e^{-τ²(m+1)}.

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Anomalous scaling limit of a Brownian particle in a log-correlated potential