Limit theorems for the one-dimensional parabolic Anderson model with white noise potential
Abstract
We consider the parabolic Anderson model ∂t u=∂x2 u+ξu on ℝ+×ℝ with u(0,·)≡ 1, where ξ is a spatial white noise. We study the long-time behavior of the spatial integral U(t):=∫−L(t)/2L(t)/2u(t,x) dx, where L(t)=exp(α3 t3/24) with α>0. We establish a weak law of large numbers for α>1 and a central limit theorem for α>2. Moreover, for every α∈(0,2), we show that U(t) converges in distribution, after explicit centering and scaling, to a totally asymmetric α-stable law. To the best of our knowledge, these are the first stable limit laws for the continuous parabolic Anderson model. In addition, we establish two spectral results of independent interest for the one-dimensional Anderson Hamiltonian on a growing interval. They are the main ingredients of the proofs. The first gives the lower-tail asymptotics of the lowest eigenvalue, including the exact prefactor. The second shows that, on this lower-tail event, the L1 norm of the corresponding eigenfunction concentrates around an explicit deterministic value.
Related
- Full asymptotics for the parabolic Anderson model with Pareto potential in the heaviest-tailed regime
- Central limit theorems for high-dimensional ridge regression under uniform subsampling
- Limit theorems for one-dimensional diffusions: reflected, skew, sticky, skew-sticky, and snapping-out limits
- Optimal dimension growth rates in the high dimensional Central Limit Theorems
- Limit theorems for Coulomb gases on a Jordan curve in an external potential
BibTeX
@misc{kim2026limit,
title = {{Limit theorems for the one-dimensional parabolic Anderson model with white noise potential}},
author = {Kunwoo Kim and Uijun Kim and Jaeyun Yi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19971},
eprint = {2609.19971},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19971},
}
RIS
TY - UNPB
TI - Limit theorems for the one-dimensional parabolic Anderson model with white noise potential
AU - Kunwoo Kim
AU - Uijun Kim
AU - Jaeyun Yi
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19971
UR - https://arxiv.org/abs/2609.19971
AB - We consider the parabolic Anderson model ∂_t u=∂_x^2 u+ξu on ℝ_+×ℝ with u(0,·)≡ 1, where ξ is a spatial white noise. We study the long-time behavior of the spatial integral U(t):=∫_−L(t)/2^L(t)/2u(t,x) dx, where L(t)=exp(α^3 t^3/24) with α>0. We establish a weak law of large numbers for α>1 and a central limit theorem for α>2. Moreover, for every α∈(0,2), we show that U(t) converges in distribution, after explicit centering and scaling, to a totally asymmetric α-stable law. To the best of our knowledge, these are the first stable limit laws for the continuous parabolic Anderson model. In addition, we establish two spectral results of independent interest for the one-dimensional Anderson Hamiltonian on a growing interval. They are the main ingredients of the proofs. The first gives the lower-tail asymptotics of the lowest eigenvalue, including the exact prefactor. The second shows that, on this lower-tail event, the L^1 norm of the corresponding eigenfunction concentrates around an explicit deterministic value.
ER -