Superdiffusion and anomalous regularization in self-similar random incompressible flows

TL;DR

The article rigorously analyzes a Brownian particle in R^d driven by a stationary, incompressible, multiscale drift encoded by a random stream matrix with positive Hurst γ. It establishes quenched power-law superdiffusivity in the perturbative regime γ≪1, confirming RG-type predictions that the displacement variance scales as t^{2/(2−γ)} with a precisely controlled random prefactor, by a Wilsonian RG treatment of the generator L=∇·(νI_d+ k)∇. A central achievement is the scale-by-scale coarse-graining, yielding a running effective diffusivity s_m with s_m≈(ν^2+c_* γ^{-1}3^{2γ m})^{1/2} and a scale-local defect E(m)=O(γ^{1/2}|log γ|^2), which allows a scale-local harmonic-approximation and an anomalous regularization bound: for almost every realization, solutions of −∇·a∇u=0 are Hölder with exponent α=1−Cγ^{1/2}, uniformly in ν. The approach blends deterministic multiscale coarse-graining with probabilistic RG flow, including sensitivity estimates, percolation-type control of bad scales, and excess-decay iterations, to propagate regularity and quantify the multifractal environment. The results connect to turbulence phenomenology via multifractal RG dynamics, provide sharp quenched exponents for superdiffusivity, and establish scale-local regularity that remains robust as ν→0, highlighting the intricate interplay between self-similarity, ergodicity, and transport in random incompressible flows.

Abstract

We study the long-time behavior of a particle in $\mathbb{R}^d$, $d \geq 2$, subject to molecular diffusion and advection by a random incompressible flow. The velocity field is the divergence of a stationary random stream matrix $\mathbf{k} $ with positive Hurst exponent $γ> 0$, so the resulting random environment is multiscale and self-similar. In the perturbative regime $γ\ll 1$, we prove quenched power-law superdiffusion: for a typical realization of the environment, the displacement variance at time $t$ grows like $t^{2/(2-γ)}$, the scaling predicted by renormalization group heuristics. We also identify the leading prefactor up to a random (quenched) relative error of order $γ^{\frac12}\left| \log γ\right|^3$. The proof implements a Wilsonian renormalization group scheme at the level of the infinitesimal generator $\nabla \cdot (νI_d + \mathbf{k} ) \nabla$, based on a self-similar induction across scales. We demonstrate that the coarse-grained generator is well-approximated, at each scale $r$, by a constant-coefficient Laplacian with effective diffusivity growing like $r^γ$. This approximation is inherently scale-local: reflecting the multifractal nature of the environment, the relative error does not decay with the scale, but remains of order $γ^{\frac12}\left| \log γ\right|^2$. We also prove anomalous regularization under the quenched law: for almost every realization of the drift, solutions of the associated elliptic equation are Hölder continuous with exponent $1 - Cγ^{\frac12}$ and satisfy estimates which are uniform in the molecular diffusivity $ν$ and the scale.

Theorems & Definitions (20)

• Theorem A: Quenched power-law superdiffusivity
• Theorem B: Renormalization of the generator
• Theorem C: Anomalous Hölder regularity
• proof : Proof of \ref{['e.shaking.lambda']}
• Lemma 2.1
• proof
• Definition 2.2: Coarse-grained ellipticity constants
• Definition 2.3: Homogenization error
• Lemma 2.4
• proof