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Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift
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abstract
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.
Forward citations
Cited by 3 Pith papers
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Effective Lagrangian regularity and the uniqueness threshold for random H\"older velocity fields
For random multiscale Hölder velocity fields with finite-range dependence, uniqueness of ODE and transport solutions holds almost surely above the sharp threshold alpha=1/2, with explicit counterexamples below.
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Construction of the 1d Self-repelling Brownian Polymer
The 1d self-repelling Brownian polymer, with the singular Dirac interaction, is constructed in the annealed random-environment setting as a unique energy solution, and it is shown to be superdiffusive.
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Quantitative estimates for high-contrast random media
For elliptic equations with random well-separated holes, the paper proves stretched-exponential tails for the regularity radius and sublinear corrector growth under a multiscale spectral gap condition.
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