{"title":"Stationary measure for the open KPZ equation","authors":"Ivan Corwin, Alisa Knizel","doi":"10.1002/cpa.22174","DOIUrl":null,"url":null,"abstract":"<p>We provide the first construction of stationary measures for the open KPZ equation on the spatial interval [0,1] with general inhomogeneous Neumann boundary conditions at 0 and 1 depending on real parameters <i>u</i> and <i>v</i>, respectively. When <math>\n <semantics>\n <mrow>\n <mi>u</mi>\n <mo>+</mo>\n <mi>v</mi>\n <mo>≥</mo>\n <mn>0</mn>\n </mrow>\n <annotation>$u+v\\ge 0$</annotation>\n </semantics></math>, we uniquely characterize the constructed stationary measures through their multipoint Laplace transform, which we prove is given in terms of a stochastic process that we call the continuous dual Hahn process. Our work relies on asymptotic analysis of Bryc and Wesołowski's Askey–Wilson process formulas for the open ASEP stationary measure (which in turn arise from Uchiyama, Sasamoto and Wadati's Askey-Wilson Jacobi matrix representation of Derrida et al.'s matrix product ansatz) in conjunction with Corwin and Shen's proof that open ASEP converges to open KPZ under weakly asymmetric scaling.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 4","pages":"2183-2267"},"PeriodicalIF":3.1000,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications on Pure and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22174","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We provide the first construction of stationary measures for the open KPZ equation on the spatial interval [0,1] with general inhomogeneous Neumann boundary conditions at 0 and 1 depending on real parameters u and v, respectively. When , we uniquely characterize the constructed stationary measures through their multipoint Laplace transform, which we prove is given in terms of a stochastic process that we call the continuous dual Hahn process. Our work relies on asymptotic analysis of Bryc and Wesołowski's Askey–Wilson process formulas for the open ASEP stationary measure (which in turn arise from Uchiyama, Sasamoto and Wadati's Askey-Wilson Jacobi matrix representation of Derrida et al.'s matrix product ansatz) in conjunction with Corwin and Shen's proof that open ASEP converges to open KPZ under weakly asymmetric scaling.
我们给出了空间区间[0,1]上开KPZ方程的平稳测度的第一个构造,该方程的一般非齐次Neumann边界条件分别为0和1,取决于实参数u和v。当u+v≥0$u+v\ge 0$时,我们通过它们的多点拉普拉斯变换唯一地刻画了构造的平稳测度,我们证明了它是根据一个随机过程给出的,我们称之为连续对偶Hahn过程。我们的工作依赖于Bryc和Wesołowski的开放ASEP平稳测度的Askey–Wilson过程公式的渐近分析(这反过来又源于Derrida等人的矩阵乘积ansatz的Uchiyama、Sasamoto和Wadati的Askey Wilson Jacobi矩阵表示),以及Corwin和Shen的证明,即开放ASEP在弱不对称标度下收敛于开放KPZ。