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KPZ equation from a class of nonlinear SPDEs in infinite volume

Kevin Yang

TL;DR

The paper proves that a broad class of nonlinear Ginzburg–Landau SPDEs in infinite volume, under weak-nonlinearity scaling and non-equilibrium initial data, converge to the KPZ equation on the full line. Its core method is a space-discretization leading to infinite-dimensional SDEs, together with a carefully constructed stochastic heat kernel and a stochastic Cole–Hopf transform, which together tame infinite-volume singularities. The authors establish universality results for both continuous and wedge-type initial data, including a wedge limit that yields Tracy–Widom fluctuations, and extend prior torus results to the real line while handling non-equilibrium data beyond Brownian-type initial data. This approach provides a robust bridge from nonlinear SPDEs to KPZ in infinite volume and connects to random-matrix theory through the wedge/slope universality, with potential implications for non-equilibrium statistical mechanics and integrable probability.

Abstract

We study a general class of nonlinear Ginzburg-Landau SPDEs in infinite volume under weak nonlinearity scaling and with non-equilibrium initial data. We derive the KPZ equation as a continuum limit of these equations. This makes rigorous the original derivation of the KPZ equation from physics in the full-space setting, which was a problem posed by Hairer-Quastel '18. Our analysis is based on a stochastic heat kernel for a linearization of said SPDEs.

KPZ equation from a class of nonlinear SPDEs in infinite volume

TL;DR

The paper proves that a broad class of nonlinear Ginzburg–Landau SPDEs in infinite volume, under weak-nonlinearity scaling and non-equilibrium initial data, converge to the KPZ equation on the full line. Its core method is a space-discretization leading to infinite-dimensional SDEs, together with a carefully constructed stochastic heat kernel and a stochastic Cole–Hopf transform, which together tame infinite-volume singularities. The authors establish universality results for both continuous and wedge-type initial data, including a wedge limit that yields Tracy–Widom fluctuations, and extend prior torus results to the real line while handling non-equilibrium data beyond Brownian-type initial data. This approach provides a robust bridge from nonlinear SPDEs to KPZ in infinite volume and connects to random-matrix theory through the wedge/slope universality, with potential implications for non-equilibrium statistical mechanics and integrable probability.

Abstract

We study a general class of nonlinear Ginzburg-Landau SPDEs in infinite volume under weak nonlinearity scaling and with non-equilibrium initial data. We derive the KPZ equation as a continuum limit of these equations. This makes rigorous the original derivation of the KPZ equation from physics in the full-space setting, which was a problem posed by Hairer-Quastel '18. Our analysis is based on a stochastic heat kernel for a linearization of said SPDEs.

Paper Structure

This paper contains 52 sections, 34 theorems, 287 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Acknowledgements
  4. Main results
  5. Precise statement of main results
  6. Universality for continuous initial data
  7. Universality for wedge-type initial data
  8. Plan for the paper
  9. Notation
  10. Heuristics behind Theorems \ref{['theorem:main']} and \ref{['theorem:mainwedge']}
  11. Proof outlines for the main results (Theorems \ref{['theorem:main']} and \ref{['theorem:mainwedge']})
  12. Step 1 -- the basic sde
  13. Step 2 -- stochastic heat kernel
  14. Step 3 -- analysis in the "almost compact" setting
  15. Proofs of the main theorems
  16. ...and 37 more sections

Key Result

Theorem 2.6

Suppose that in addition to Assumptions assump:potential and assump:noneq, the following assumptions are satisfied. Then, there exists a coupling between $\mathbf{Z}^{N}$ and the solution $\mathbf{Z}$ to eq:she with initial data $\mathbf{Z}_{0,\cdot}$ from bullet point (2) above such that $\mathbf{Z}^{N}_{{t},N{X}}-\mathbf{Z}_{{t},{X}}\to0$ locally uniformly on $[0,1]\times\mathbb R$ in probabili

Theorems & Definitions (62)

  • Definition 2.1
  • Remark 2.3
  • Example 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Remark 3.1
  • Remark 3.2
  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • ...and 52 more