Hamilton-Jacobi equations from mean-field spin glasses

• 成果类型: Article
• 署名作者: Chen, Hong-Bin; Xia, Jiaming
• 署名单位: New York University; University of Pennsylvania
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01386-5
• 发表日期: 2025
• 页码: 803-873
• 关键词: viscosity solutions; free-energy; infinite dimensions; MODEL; FORMULA

摘要

We give a meaning to the Hamilton-Jacobi equation arising from mean-field spin glass models in the viscosity sense, and establish the corresponding well-posedness. Originally defined on the set of monotone probability measures, these equations can be interpreted, via an isometry, to be defined on an infinite-dimensional closed convex cone with an empty interior in a Hilbert space. We prove the comparison principle, and the convergence of finite-dimensional approximations furnishing the existence of solutions. Under additional convexity conditions, we show that the solution can be represented by a version of the Hopf-Lax formula, or the Hopf formula on cones. Previously, two notions of solutions were considered, one defined directly as the Hopf-Lax formula, and another as limits of finite-dimensional approximations. They have been proven to describe the limit free energy in a wide class of mean-field spin glass models. This work shows that these two kinds of solutions are viscosity solutions.