42. chen.jimenez:26:spatial
^^^^^^^^^^^^^^^^^^^^^^^^^^

.. centered:: Spatial covariance of KPZ from flat initial profile
.. centered:: Le Chen and Juan J. Jiménez

**Abstract:** We study the fixed-time spatial covariance of the KPZ equation
with flat initial profile. Using Malliavin calculus and a Clark-Ocone
representation, we show that as :math:`|x|\to\infty`,
:math:`\mathrm{Cov}[h(t,x),h(t,0)]` is governed by a boundary-layer regime
near the initial time and satisfies

.. math::

   \mathrm{Cov}[h(t,x),h(t,0)] \sim \kappa(t) \int_0^t p_{2r}(x)\,dr
   = \frac{2\kappa(t)}{\sqrt{\pi}}\,t^{3/2}\,|x|^{-2}
   \exp\!\left(-\frac{x^2}{4t}\right),

as :math:`|x|\to\infty`, where
:math:`\kappa(t) = \big(\mathbb{E}[Z(t,0)^{-1}]\big)^2`, :math:`Z` is the
flat stochastic heat equation solution, and :math:`p_t` is the one-dimensional
heat kernel. In sharp contrast with the narrow-wedge regime, where Gu--Pu
(2025, Theorem 1.1) proved that for each fixed :math:`t>0`,

.. math::

   \mathrm{Cov}\!\left[h^{\mathrm{nw}}(t,x),h^{\mathrm{nw}}(t,0)\right]
   \sim \frac{t}{|x|},

as :math:`|x|\to\infty`, the flat initial profile exhibits Gaussian decay,
yielding, to the best of our knowledge, the first exact spatial covariance
asymptotic for the KPZ equation under flat initial data. We also establish an
explicit closed-form formula for the second moment of the continuum directed
random polymer partition function.

**MSC 2020 subject classifications:** 60H15, 60H07, 35R60.

**Keywords:** KPZ equation, spatial covariance, flat initial profile, Malliavin calculus, Clark-Ocone formula, directed random polymer.

`Preprint <https://arxiv.org/abs/2603.14174>`_

:cite:`chen.jimenez:26:spatial` Le Chen & Juan J. Jiménez (2026) 'Spatial
covariance of KPZ from flat initial profile', *preprint arXiv:2603.14174, 25
pages*

.. code-block:: bibtex

   @article{chen.jimenez:26:spatial,
      title         = {Spatial covariance of KPZ from flat initial profile},
      author        = {Le Chen and Juan J. Jim{\'e}nez},
      year          = {2026},
      month         = {March},
      journal       = {Preprint arXiv:2603.14174},
      url           = {http://arXiv.org/abs/2603.14174}
   }


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