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Spatial covariance of KPZ from flat initial profile

Le Chen, 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 $|x|\to\infty$, $\mathrm{Cov}[h(t,x),h(t,0)]$ is governed by a boundary-layer regime near the initial time and satisfies $\mathrm{Cov}[h(t,x),h(t,0)] \sim κ(t) \int_0^t p_{2r}(x) dr = \frac{2κ(t)}{\sqrtπ}t^{3/2}|x|^{-2}\exp\left(-\frac{x^2}{4t}\right),$ as $|x|\to\infty$, where $κ(t) = \big(\mathbb{E}[Z(t,0)^{-1}]\big)^2$, $Z$ is the flat stochastic heat equation solution, and $p_t$ 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 $t>0$, $\mathrm{Cov}\left[h^{\mathrm{nw}}(t,x),h^{\mathrm{nw}}(t,0)\right]\sim \frac{t}{|x|},$ as $|x|\to\infty$, 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.

Spatial covariance of KPZ from flat initial profile

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 , is governed by a boundary-layer regime near the initial time and satisfies as , where , is the flat stochastic heat equation solution, and 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 , as , 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.
Paper Structure (18 sections, 24 theorems, 153 equations)

This paper contains 18 sections, 24 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Clark--Ocone representation and basic bounds
  4. Clark--Ocone covariance representation
  5. Known integrability and a Gaussian-tail upper bound
  6. Identifying the covariance constant.
  7. Moment and inverse-moment bounds
  8. Continuum polymer.
  9. Shear mixing for space--time white noise
  10. Step I: weak convergence of the shear operators.
  11. Step II: the case of the $n$-th Wiener chaos.
  12. Step III: finite chaos expansions.
  13. Step IV: $L^2$ approximation.
  14. Spatial decay: boundary layer approach
  15. Boundary-layer reduction
  16. ...and 3 more sections

Key Result

Theorem 1.1

For flat initial data E:flat-initial, for all $t>0$, it holds that

Theorems & Definitions (54)

  • Theorem 1.1: Flat covariance asymptotic
  • Remark 1.2: Strategy and main difficulties
  • Theorem 1.3: Explicit second moment of $\bar{\mathcal{G}}_\beta$
  • Proposition 2.1: Flat initial: Gaussian-tail covariance upper bound
  • proof
  • Lemma 2.2: Removing $Z(s,y)^2$ from the boundary layer
  • proof
  • Remark 3.1
  • proof : Proof of Theorem \ref{['thm:barG-second-moment-intro']}
  • Remark 3.2: Comparison with the parabolic Anderson model
  • ...and 44 more