48. balan.chen.ea:26:functional

A functional central limit theorem for the KPZ equation with narrow wedge initial data

Raluca M. Balan, Le Chen, and Jinxin Wang

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.

The main result: with \(S_N(t)\) the spatial integral over \([-N,N]\) of the centered logarithm of the normalized narrow-wedge solution,

\[\frac{1}{\sqrt{N\log N}}\, S_N(\cdot) \xrightarrow{\ d\ } \sqrt{2}\, B(\cdot) \quad \text{in } C[\varepsilon,T], \qquad 0<\varepsilon<T,\]

where \(B\) is a standard Brownian motion.

Keywords: KPZ equation; narrow wedge initial data; functional central limit theorem; spatial average; Malliavin calculus; Clark-Ocone formula. MSC: Primary 60H15; Secondary 60F05, 60H07, 35R60.

Preprint

[BCW26] Raluca M. Balan, Le Chen & Jinxin Wang (2026) ‘A functional central limit theorem for the KPZ equation with narrow wedge initial data’, preprint arXiv:2610.27219, 25 pages

@article{balan.chen.ea:26:functional,
   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},
   year          = {2026},
   month         = {October},
   journal       = {Preprint arXiv:2610.27219},
   url           = {http://arXiv.org/abs/2610.27219}
}

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