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Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift

Scott Armstrong, Ahmed Bou-Rabee, Tuomo Kuusi

TL;DR

This work analyzes a Brownian particle in R^d advected by a stationary, divergence-free, log-correlated random field, establishing a quenched superdiffusive central limit theorem with variance growing as $2 d c_* t (\log t)^{1/2}$ and a quenched invariance principle under a log-scale renormalization. The authors develop a rigorous multiscale renormalization group framework built on quantitative homogenization of the generator $\mathfrak{L}=\nabla\cdot(\nu I_d+\mathbf{k})\nabla$, introducing coarse-grained diffusion matrices and infrared cutoffs to track scale-by-scale diffusivity enhancements. They prove large-scale regularity results (Liouville theorems and $C^{0,\gamma}$/$C^{1,\gamma}$ estimates) and a detailed, scale-guided proof that the effective diffusivity grows like $\sqrt{2 c_* \log 3^m}$, culminating in a quenched invariance principle via homogenization of the infinitesimal generator. The paper combines localization, sharp coarse-graining, perturbative analysis, and stochastic concentration to achieve a full, quenched, multiscale picture of critical-log-correlated advection and demonstrates new methods in iterative homogenization applicable to other critical phenomena.

Abstract

We consider the long-time behavior of a diffusion process on $\mathbb{R}^d$ advected by a stationary random vector field which is assumed to be divergence-free, dihedrally symmetric in law and have a log-correlated potential. A special case includes $\nabla^\perp$ of the Gaussian free field in two dimensions. We show the variance of the diffusion process at a large time $t$ behaves like $2 c_* t (\log t)^{1/2}$, in a quenched sense and with a precisely determined, universal prefactor constant $c_*>0$. We also prove a quenched invariance principle under this superdiffusive scaling. The proof is based on a rigorous renormalization group argument in which we inductively analyze coarse-grained diffusivities, scale-by-scale. Our analysis leads to sharp homogenization and large-scale regularity estimates on the infinitesimal generator, which are subsequently transferred into quantitative information on the process.

Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift

TL;DR

This work analyzes a Brownian particle in R^d advected by a stationary, divergence-free, log-correlated random field, establishing a quenched superdiffusive central limit theorem with variance growing as and a quenched invariance principle under a log-scale renormalization. The authors develop a rigorous multiscale renormalization group framework built on quantitative homogenization of the generator , introducing coarse-grained diffusion matrices and infrared cutoffs to track scale-by-scale diffusivity enhancements. They prove large-scale regularity results (Liouville theorems and / estimates) and a detailed, scale-guided proof that the effective diffusivity grows like , culminating in a quenched invariance principle via homogenization of the infinitesimal generator. The paper combines localization, sharp coarse-graining, perturbative analysis, and stochastic concentration to achieve a full, quenched, multiscale picture of critical-log-correlated advection and demonstrates new methods in iterative homogenization applicable to other critical phenomena.

Abstract

We consider the long-time behavior of a diffusion process on advected by a stationary random vector field which is assumed to be divergence-free, dihedrally symmetric in law and have a log-correlated potential. A special case includes of the Gaussian free field in two dimensions. We show the variance of the diffusion process at a large time behaves like , in a quenched sense and with a precisely determined, universal prefactor constant . We also prove a quenched invariance principle under this superdiffusive scaling. The proof is based on a rigorous renormalization group argument in which we inductively analyze coarse-grained diffusivities, scale-by-scale. Our analysis leads to sharp homogenization and large-scale regularity estimates on the infinitesimal generator, which are subsequently transferred into quantitative information on the process.
Paper Structure (44 sections, 84 theorems, 1198 equations, 2 figures)

This paper contains 44 sections, 84 theorems, 1198 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Superdiffusive central limit theorem
  3. Quantitative homogenization
  4. Liouville theorems and large-scale regularity
  5. Heuristic arguments for the superdiffusive scaling
  6. An outline of the rigorous proof
  7. Notation
  8. Coarse-graining estimates
  9. Definition of the coarse-grained matrices
  10. Basic properties of the coarse-grained matrices
  11. Orlicz notation for tail bounds
  12. Infrared cutoffs
  13. Localization
  14. A suboptimal lower bound on the renormalized diffusivities
  15. Homogenization on scales below the infrared cutoff
  16. ...and 29 more sections

Key Result

Theorem A

There exists a constant $c_*(\mathbb{P})>0$ such that, for $\mathbb{P}$--a.e. realization of the vector field $\mathbf{f}$, where $\{ W_t \}$ is a standard Brownian motion on $\mathbb{R}^d$ and the convergence in e.convinlaw is in law, with respect to the uniform topology on paths. Moreover, for each $\delta\in (0,1/4)$ and $\beta \in (0,4\delta)$, there exists a constant $C(\beta,\delta,c_*,\nu,

Figures (2)

  • Figure 1.1: A Brownian motion subject to a random incompressible drift.
  • Figure 1.2: Outline of the proof.

Theorems & Definitions (170)

  • Theorem A: Quenched superdiffusive invariance principle
  • Theorem B: Quantitative homogenization
  • Theorem C: Large-scale $C^{0,\gamma}$ estimate
  • Theorem D: Large-scale $C^{1,\gamma}$ estimate
  • Conjecture E
  • Conjecture F
  • Lemma 2.1: Generalized triangle inequality
  • proof
  • Lemma 2.2: Multiplication property
  • proof
  • ...and 160 more