Jointly invariant measures for the Kardar-Parisi-Zhang equation

• 成果类型: Article
• 署名作者: Groathouse, Sean; Rassoul-Agha, Firas; Seppalainen, Timo; Sorensen, Evan
• 署名单位: Utah System of Higher Education; University of Utah; University of Wisconsin System; University of Wisconsin Madison; Columbia University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01362-z
• 发表日期: 2025
• 页码: 303-372
• 关键词: stochastic heat-equation; hamilton-jacobi equations; burgers-equation; directed polymers; stationary solutions; limit; distributions; BEHAVIOR

摘要

We give an explicit description of a family of jointly invariant measures of the KPZ equation singled out by asymptotic slope conditions. These are couplings of Brownian motions with drift, and can be extended to a cadlag process indexed by all real drift parameters. We name this process the KPZ horizon (KPZH). As a corollary, we resolve a recent conjecture by showing the existence of a random, countably infinite dense set of drift values at which the Busemann process of the KPZ equation is discontinuous. This signals instability, and shows the failure of the one force-one solution principle and the existence of at least two extremal semi-infinite polymer measures in the exceptional directions. The low-temperature limit of the KPZH is the stationary horizon (SH), the unique jointly invariant measure of the KPZ fixed point under the same slope conditions. The high-temperature limit of the KPZH is a coupling of Brownian motions that differ by linear shifts, which is jointly invariant under the Edwards-Wilkinson fixed point.