开放KPZ方程的不变测度：解析视角
Invariant measures for the open KPZ equation: an analytic perspective
AI总结：
本文通过随机分析方法，证明了开放KPZ方程的不变测度公式，利用高斯测度、测度变换和正则结构理论，建立了边界层内的中心极限定理。
摘要：
开放KPZ方程的遍历理论近年来取得了显著进展，一系列由Corwin-Knizel、Barraquand-Le Doussal以及Bryc-Kuznetsov-Wang-Wesołowski等人的工作中对显式不变测度进行了描述。本文提供了一个随机分析的证明，用于证明不变测度的公式。我们的方法从同质边界条件下的高斯不变测度开始。通过将非齐次问题近似为带有奇异边界势的同质问题，利用测度变换、马尔可夫过程的时间反转以及伊藤公式等工具，我们将问题归结为分析边界层内的KPZ非线性项。最后，利用正则结构理论，我们建立了边界附近时间积分非线性项的中心极限定理，从而完成了不变性的证明。尽管已知不同的边界参数会导致不变测度的不同物理区域，但我们的方法具有鲁棒性，不依赖于任何特定的边界参数选择。
英文摘要：
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.
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