Fokker-Planck equations on discrete infinite graphs
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917083080556544 |
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| author | Carrillo, Jose A. Wang, Xinyu |
| author_facet | Carrillo, Jose A. Wang, Xinyu |
| contents | We study the gradient flow structure and long-time behavior of Fokker-Planck equations (FPE) on infinite graphs, along with a Talagrand-type inequality in this setting. We begin by constructing an infinite-dimensional Hilbert manifold structure, extending the approach of [S. N. Chow, W. Huang, Y. Li, H. M. Zhou, Arch. Ration. Mech. Anal., 203, 969-1008 (2012)] through a novel classification method to establish injectivity of the map from quotient space to tangent space and employing functional analysis techniques to prove surjectivity. Using a combination of the relative energy method, approximation techniques, and continuity arguments, we establish the global existence and asymptotic convergence of solutions to the infinite-dimensional ODE system associated with the FPE. Specifically, we demonstrate that the FPE admits a gradient flow structure, with solutions converging exponentially to the unique Gibbs distribution. Furthermore, we prove a local Talagrand-type inequality and compare the Hilbert manifold metric induced by our framework with classical Wasserstein distances. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_12076 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fokker-Planck equations on discrete infinite graphs Carrillo, Jose A. Wang, Xinyu Analysis of PDEs We study the gradient flow structure and long-time behavior of Fokker-Planck equations (FPE) on infinite graphs, along with a Talagrand-type inequality in this setting. We begin by constructing an infinite-dimensional Hilbert manifold structure, extending the approach of [S. N. Chow, W. Huang, Y. Li, H. M. Zhou, Arch. Ration. Mech. Anal., 203, 969-1008 (2012)] through a novel classification method to establish injectivity of the map from quotient space to tangent space and employing functional analysis techniques to prove surjectivity. Using a combination of the relative energy method, approximation techniques, and continuity arguments, we establish the global existence and asymptotic convergence of solutions to the infinite-dimensional ODE system associated with the FPE. Specifically, we demonstrate that the FPE admits a gradient flow structure, with solutions converging exponentially to the unique Gibbs distribution. Furthermore, we prove a local Talagrand-type inequality and compare the Hilbert manifold metric induced by our framework with classical Wasserstein distances. |
| title | Fokker-Planck equations on discrete infinite graphs |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.12076 |