48. balan.chen.ea:26:functional
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.. centered:: A functional central limit theorem for the KPZ equation with narrow wedge initial data
.. centered:: 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 :math:`S_N(t)` the spatial integral over :math:`[-N,N]`
of the centered logarithm of the normalized narrow-wedge solution,

.. math::

   \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 :math:`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 <https://arxiv.org/abs/2610.27219>`_

:cite:`balan.chen.ea:26:functional` 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*

.. code-block:: bibtex

   @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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