Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift
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arXiv
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| Format: | Preprint |
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2024
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| author | Armstrong, Scott Bou-Rabee, Ahmed Kuusi, Tuomo |
| author_facet | Armstrong, Scott Bou-Rabee, Ahmed Kuusi, Tuomo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01115 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift Armstrong, Scott Bou-Rabee, Ahmed Kuusi, Tuomo Probability Mathematical Physics Analysis of PDEs 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. |
| title | Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift |
| topic | Probability Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2404.01115 |