Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2022
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2201.12732 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866916807728693248 |
|---|---|
| author | Chen, Hong-Bin Xia, Jiaming |
| author_facet | Chen, Hong-Bin Xia, Jiaming |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_12732 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hamilton-Jacobi equations from mean-field spin glasses Chen, Hong-Bin Xia, Jiaming Analysis of PDEs Disordered Systems and Neural Networks Probability 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. |
| title | Hamilton-Jacobi equations from mean-field spin glasses |
| topic | Analysis of PDEs Disordered Systems and Neural Networks Probability |
| url | https://arxiv.org/abs/2201.12732 |