Superdiffusion and anomalous regularization in self-similar random incompressible flows
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917232970301440 |
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| author | Armstrong, Scott Bou-Rabee, Ahmed Kuusi, Tuomo |
| author_facet | Armstrong, Scott Bou-Rabee, Ahmed Kuusi, Tuomo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_22142 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Superdiffusion and anomalous regularization in self-similar random incompressible flows Armstrong, Scott Bou-Rabee, Ahmed Kuusi, Tuomo Probability Mathematical Physics Analysis of PDEs 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. |
| title | Superdiffusion and anomalous regularization in self-similar random incompressible flows |
| topic | Probability Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2601.22142 |