Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Invariant measures for the open KPZ equation: an analytic perspective

Alexander Dunlap, Yu Gu, Tommaso Rosati

TL;DR

This work delivers a stochastic analytic proof of the open KPZ invariant-measure description on a finite strip by starting from the Gaussian invariant measure at homogeneous boundaries and controlled singular boundary perturbations. The authors synthesize a four-pronged strategy: time-reversal symmetry for the zero-boundary case, a Cameron–Martin change of measure to encode boundary data, Itô’s formula to expose a martingale structure for the Radon–Nikodym factor, and a boundary-adapted regularity-structure analysis to tame the singular boundary nonlinearity. Central results include a precise boundary-layer description, a central-limit-type behavior for the time-integrated boundary flux, and a rigorous derivation of the Radon–Nikodym derivative that characterizes the invariant measure for all boundary parameters. The approach is robust and does not rely on discrete integrable approximations, offering a direct analytic pathway to understanding boundary effects and invariant measures in open KPZ. The work thereby advances both the mathematical theory of singular SPDEs with boundary conditions and the physical interpretation of boundary phenomena in KPZ-type systems.

Abstract

The ergodic theory of the open KPZ equation has seen significant progress in recent years, with explicit invariant measures described in a series of works by Corwin--Knizel, Barraquand--Le Doussal, and Bryc--Kuznetsov--Wang--Wesołowski. In this paper, we provide a stochastic analytic proof of the formula for the invariant measures. Our approach starts from the Gaussian invariant measure for the case of homogeneous boundary conditions. We approximate the inhomogeneous problem by a homogeneous one with a singular boundary potential. Using tools including change of measure, time reversal for Markov processes, and Itô's formula, we then reduce the problem to analyzing the KPZ nonlinearity in a thin boundary layer. Finally, using the theory of regularity structures, we establish a central limit theorem for the time-integrated nonlinearity near the boundary, which completes the proof of the invariance. Although it is known that different boundary parameters give rise to distinct physical regimes for the invariant measures, our method is robust and does not rely on any particular choice of boundary parameters.

Invariant measures for the open KPZ equation: an analytic perspective

TL;DR

This work delivers a stochastic analytic proof of the open KPZ invariant-measure description on a finite strip by starting from the Gaussian invariant measure at homogeneous boundaries and controlled singular boundary perturbations. The authors synthesize a four-pronged strategy: time-reversal symmetry for the zero-boundary case, a Cameron–Martin change of measure to encode boundary data, Itô’s formula to expose a martingale structure for the Radon–Nikodym factor, and a boundary-adapted regularity-structure analysis to tame the singular boundary nonlinearity. Central results include a precise boundary-layer description, a central-limit-type behavior for the time-integrated boundary flux, and a rigorous derivation of the Radon–Nikodym derivative that characterizes the invariant measure for all boundary parameters. The approach is robust and does not rely on discrete integrable approximations, offering a direct analytic pathway to understanding boundary effects and invariant measures in open KPZ. The work thereby advances both the mathematical theory of singular SPDEs with boundary conditions and the physical interpretation of boundary phenomena in KPZ-type systems.

Abstract

The ergodic theory of the open KPZ equation has seen significant progress in recent years, with explicit invariant measures described in a series of works by Corwin--Knizel, Barraquand--Le Doussal, and Bryc--Kuznetsov--Wang--Wesołowski. In this paper, we provide a stochastic analytic proof of the formula for the invariant measures. Our approach starts from the Gaussian invariant measure for the case of homogeneous boundary conditions. We approximate the inhomogeneous problem by a homogeneous one with a singular boundary potential. Using tools including change of measure, time reversal for Markov processes, and Itô's formula, we then reduce the problem to analyzing the KPZ nonlinearity in a thin boundary layer. Finally, using the theory of regularity structures, we establish a central limit theorem for the time-integrated nonlinearity near the boundary, which completes the proof of the invariance. Although it is known that different boundary parameters give rise to distinct physical regimes for the invariant measures, our method is robust and does not rely on any particular choice of boundary parameters.

Paper Structure

This paper contains 73 sections, 74 theorems, 785 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Our method
  3. A toy example
  4. Key steps in the proof
  5. Related work
  6. Outline of the paper
  7. Notation
  8. Acknowledgments
  9. Equations, time reversal, and approximations
  10. Mild and energy solutions
  11. Time reversal
  12. The boundary potential approximation and change of measure
  13. Equations with boundary potentials
  14. Cameron--Martin changes of measure
  15. Noise mollification
  16. ...and 58 more sections

Key Result

Theorem 1.1

Let $\mathsf{u},\mathsf{v}\in\mathbb{R}$. The invariant measure $\mu_{\mathsf{u},\mathsf{v}}$ for []eq:uuv[]eq:uuv is absolutely continuous with respect to the law $\mu_{0,0}$ of spatial white noise on $[0,L]$ with Radon--Nikodym derivative given in []eq:Duv-def[]eq:Duv-def.

Theorems & Definitions (156)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Proposition 2.5
  • proof
  • ...and 146 more