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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2412.20610 |
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| _version_ | 1866914028296601600 |
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| author | Chen, Hong-Bin |
| author_facet | Chen, Hong-Bin |
| contents | Recently, [arXiv:2311.08980] demonstrated that, if it exists, the limit free energy of possibly non-convex spin glass models must be determined by a characteristic of the associated infinite-dimensional non-convex Hamilton-Jacobi equation. In this work, we investigate a similar theme purely from the perspective of PDEs. Specifically, we study the unique viscosity solution of the aforementioned equation and derive an envelope-type representation formula for the solution, in the form proposed by Evans in [doi:10.1007/s00526-013-0635-3]. The value of the solution is expressed as an average of the values along characteristic lines, weighted by a non-explicit probability measure. The technical challenges arise not only from the infinite dimensionality but also from the fact that the equation is defined on a closed convex cone with an empty interior, rather than on the entire space. In the introduction, we provide a description of the motivation from spin glass theory and present the corresponding results for comparison with the PDE results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20610 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Envelope representation of Hamilton-Jacobi equations from spin glasses Chen, Hong-Bin Analysis of PDEs Disordered Systems and Neural Networks 35F21, 49L25, 82B44 Recently, [arXiv:2311.08980] demonstrated that, if it exists, the limit free energy of possibly non-convex spin glass models must be determined by a characteristic of the associated infinite-dimensional non-convex Hamilton-Jacobi equation. In this work, we investigate a similar theme purely from the perspective of PDEs. Specifically, we study the unique viscosity solution of the aforementioned equation and derive an envelope-type representation formula for the solution, in the form proposed by Evans in [doi:10.1007/s00526-013-0635-3]. The value of the solution is expressed as an average of the values along characteristic lines, weighted by a non-explicit probability measure. The technical challenges arise not only from the infinite dimensionality but also from the fact that the equation is defined on a closed convex cone with an empty interior, rather than on the entire space. In the introduction, we provide a description of the motivation from spin glass theory and present the corresponding results for comparison with the PDE results. |
| title | Envelope representation of Hamilton-Jacobi equations from spin glasses |
| topic | Analysis of PDEs Disordered Systems and Neural Networks 35F21, 49L25, 82B44 |
| url | https://arxiv.org/abs/2412.20610 |