Superdiffusion and anomalous regularization in self-similar 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 $\gamma > 0$, so the resulting random environment is multiscale and self-similar. In the perturbative regime $\gamma \ll 1$, we prove quenched power-law superdiffusion: for a typical realization of the environment, the displacement variance at time $t$ grows like $t^{2/(2-\gamma)}$, the scaling predicted by renormalization group heuristics. We also identify the leading prefactor up to a random (quenched) relative error of order $\gamma^{\frac12}\left| \log \gamma \right|^3$. The proof implements a Wilsonian renormalization group scheme at the level of the infinitesimal generator $\nabla \cdot (\nu I_d + \mathbf{k} ) \nabla$, 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 $r^\gamma$. 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 $\gamma^{\frac12}\left| \log \gamma \right|^2$. We also prove anomalous regularization under the quenched law: for almost every realization of the drift, solutions of the associated elliptic equation are H\"older continuous with exponent $1 - C\gamma^{\frac12}$ and satisfy estimates which are uniform in the molecular diffusivity $\nu$ and the scale.