Superdiffusion and anomalous regularization in self-similar random incompressible flows

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.

💡 Research Summary

The paper investigates the long‑time behavior of a passive tracer moving in a random incompressible flow in dimensions d ≥ 2, where the drift is given by the divergence of a stationary random stream matrix k with a positive Hurst exponent γ > 0. The authors work in the perturbative regime γ ≪ 1 and develop a rigorous Wilsonian renormalization‑group (RG) analysis at the level of the infinitesimal generator L = ∇·(νI_d + k)∇. Their main contributions are twofold.

First, they prove quenched super‑diffusion. For a typical realization of the random environment, the variance of the particle’s displacement satisfies \