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Stationary Measure of the Open KPZ Equation through the Enaud-Derrida Representation

Zoe Himwich

TL;DR

The paper proves that the weak limit of RN-reweighted open ASEP stationary measures, under a weak-asymmetry rescaling and the Enaud–Derrida representation, yields the open KPZ stationary measure on $[0,L]$ for $u+v>0$. By establishing pointwise convergence of the RN derivatives to the KPZ RN factor and proving finite partition functions, the authors obtain a robust probabilistic construction of the open KPZ stationary measure without relying on finite-dimensional distributions. The approach generalizes the construction to arbitrary interval length $L$ and parameter regimes in the fan region, providing an alternative route to the probabilistic formulation. The results connect open ASEP, the DEHP algebra, and Brownian reweightings to the open KPZ fixed-point, with implications for understanding stationary states in stochastic interface growth and related stochastic PDEs.

Abstract

Recent works of Barraquand and Le Doussal and Bryc, Kuznetsov, Wang, and Wesolowski gave a description of the open KPZ stationary measure as the sum of a Brownian motion and a Brownian motion reweighted by a Radon-Nikodym derivative. Subsequent work of Barraquand and Le Doussal used the Enaud-Derrida representation of the DEHP algebra to formulate the open ASEP stationary measure in terms of the sum of a random walk and a random walk reweighted by a Radon-Nikodym derivative. They show that this Radon-Nikodym derivative converges pointwise to the Radon-Nikodym derivative that characterizes the open KPZ stationary measure. This article proves that the corresponding sequence of measures converges weakly to the open KPZ stationary measure. This provides an alternative proof of the probabilistic formulation of the open KPZ stationary measure, which avoids dealing explicitly with finite dimensional distributions. We also provide the first construction of the measure on intervals of a general length and for the full range of parameters in the fan region $(u+v>0)$.

Stationary Measure of the Open KPZ Equation through the Enaud-Derrida Representation

TL;DR

The paper proves that the weak limit of RN-reweighted open ASEP stationary measures, under a weak-asymmetry rescaling and the Enaud–Derrida representation, yields the open KPZ stationary measure on for . By establishing pointwise convergence of the RN derivatives to the KPZ RN factor and proving finite partition functions, the authors obtain a robust probabilistic construction of the open KPZ stationary measure without relying on finite-dimensional distributions. The approach generalizes the construction to arbitrary interval length and parameter regimes in the fan region, providing an alternative route to the probabilistic formulation. The results connect open ASEP, the DEHP algebra, and Brownian reweightings to the open KPZ fixed-point, with implications for understanding stationary states in stochastic interface growth and related stochastic PDEs.

Abstract

Recent works of Barraquand and Le Doussal and Bryc, Kuznetsov, Wang, and Wesolowski gave a description of the open KPZ stationary measure as the sum of a Brownian motion and a Brownian motion reweighted by a Radon-Nikodym derivative. Subsequent work of Barraquand and Le Doussal used the Enaud-Derrida representation of the DEHP algebra to formulate the open ASEP stationary measure in terms of the sum of a random walk and a random walk reweighted by a Radon-Nikodym derivative. They show that this Radon-Nikodym derivative converges pointwise to the Radon-Nikodym derivative that characterizes the open KPZ stationary measure. This article proves that the corresponding sequence of measures converges weakly to the open KPZ stationary measure. This provides an alternative proof of the probabilistic formulation of the open KPZ stationary measure, which avoids dealing explicitly with finite dimensional distributions. We also provide the first construction of the measure on intervals of a general length and for the full range of parameters in the fan region .
Paper Structure (13 sections, 21 theorems, 121 equations)

This paper contains 13 sections, 21 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Outline and Discussion
  3. Acknowledgements
  4. Background and Main Technical Result
  5. Notation
  6. The open ASEP
  7. Matrix Product Ansatz Techniques and the Enaud-Derrida Representation
  8. Proof of \ref{['t:main1']}
  9. Two Key Estimates
  10. Bounds on the Path Measure
  11. Weak Convergence
  12. Technical Lemmas
  13. Pointwise Convergence

Key Result

Theorem 1.4

The stationary measure of the open KPZ increment process on the interval $[0,L]$ with Neumann boundary conditions $u,v\in\mathbb{R}$ which satisfy $u+v>0$ is given by $\{H(x)-G_{u,v}(x)\}_{x\in[0,L]}$ where $(X,G_{u,v}(x), H(x))$ are sampled from the distribution defined by $\mathbb{Q}_{L;u,v}$ (d:k

Theorems & Definitions (46)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 2.1
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5: Equation (25) BD23
  • Remark 2.6
  • ...and 36 more