Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift
Abstract
We consider the long-time behavior of a diffusion process on $\mathbb{R}^{d} $ 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 ∇ ⊥ ∇ ⊥ of the Gaussian free field in two dimensions. We show the variance and the second moment of the displacement at a large time t t behave like 2dc*nicefrac 12 t (log t)nicefrac 12 , in a quenched sense and with a precisely determined prefactor c*= c*( ℙ )>0 c ∗ = c ∗ ( P ) > 0 , independent of $\nu $ ν . 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.
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BibTeX
@article{armstrong2026superdiffusive,
title = {{Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift}},
author = {Scott N. Armstrong and Ahmed Bou‐Rabee and Tuomo Kuusi},
journal = {Inventiones mathematicae},
year = {2026},
month = oct,
doi = {10.1007/s00222-026-01455-z},
url = {https://doi.org/10.1007/s00222-026-01455-z},
note = {CC BY},
}
RIS
TY - JOUR
TI - Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift
AU - Scott N. Armstrong
AU - Ahmed Bou‐Rabee
AU - Tuomo Kuusi
PY - 2026
DA - 2026/10/05
JO - Inventiones mathematicae
DO - 10.1007/s00222-026-01455-z
UR - https://doi.org/10.1007/s00222-026-01455-z
AB - We consider the long-time behavior of a diffusion process on $\mathbb{R}^{d} $ 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 ∇ ^⊥ ∇ ⊥ of the Gaussian free field in two dimensions. We show the variance and the second moment of the displacement at a large time t t behave like 2dc_*^nicefrac 12 t (log t)^nicefrac 12 , in a quenched sense and with a precisely determined prefactor c_*= c_*( ℙ )>0 c ∗ = c ∗ ( P ) > 0 , independent of $\nu $ ν . 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.
SN - 0020-9910
SN - 1432-1297
ER -