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

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.