PreprintCC BY 4.0arXiv
A functional central limit theorem for the KPZ equation with narrow wedge initial data
Abstract
In this article, we consider the Kardar-Parisi-Zhang (KPZ) equation with narrow wedge initial data. We prove a functional central limit theorem for the spatial average of the solution, which solves an open problem from Gu and Pu (2025). Our main tool is Clark-Ocone formula from Malliavin calculus, combined with an in-depth analysis of the increments of the solution. The main step is a moment bound for time increments of the normalized spatial integral. For this, we introduce a new technique, called a ``common-ray'' change of variables. Using the fact that Malliavin derivative of the solution satisfies a Chapman-Kolmogorov relation, we reduce the kernel differences at two times to spatial increments at one common time.
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BibTeX
@misc{balan2026functional,
title = {{A functional central limit theorem for the KPZ equation with narrow wedge initial data}},
author = {Raluca M. Balan and Le Chen and Jinxin Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27219},
eprint = {2610.27219},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27219},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A functional central limit theorem for the KPZ equation with narrow wedge initial data
AU - Raluca M. Balan
AU - Le Chen
AU - Jinxin Wang
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27219
UR - https://arxiv.org/abs/2610.27219
AB - In this article, we consider the Kardar-Parisi-Zhang (KPZ) equation with narrow wedge initial data. We prove a functional central limit theorem for the spatial average of the solution, which solves an open problem from Gu and Pu (2025). Our main tool is Clark-Ocone formula from Malliavin calculus, combined with an in-depth analysis of the increments of the solution. The main step is a moment bound for time increments of the normalized spatial integral. For this, we introduce a new technique, called a ``common-ray'' change of variables. Using the fact that Malliavin derivative of the solution satisfies a Chapman-Kolmogorov relation, we reduce the kernel differences at two times to spatial increments at one common time.
ER -