Gradient flow structure, well-posedness and asymptotic behavior of Fokker-Planck equation on locally finite graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918197372911616 |
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| author | Wang, Cong |
| author_facet | Wang, Cong |
| contents | This paper investigates the gradient flow structure, well-posedness, and asymptotic behavior of the Fokker-Planck equation defined on locally uniformly finite graphs, which is highly non-trivial compared with the finite case. We first construct a 2-Wasserstein-type metric and gradient flow equation in the probability density space associated with the underlying graphs. Then, we prove the global existence of solution to the Fokker-Planck equation using a novel approach that differs significantly from the methods applied in the finite case. We also demonstrate that the solution converges to the Gibbs distribution in the $\ell^{r}(V,\bmπ)$ norm with $r\in [2,\infty]$, by using the indicator set partitioning method. To the best of our knowledge, this work seems the first result on the study of Wasserstein-type metrics and the Fokker-Planck equation in probability density spaces defined on infinite graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_03531 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gradient flow structure, well-posedness and asymptotic behavior of Fokker-Planck equation on locally finite graphs Wang, Cong Probability Analysis of PDEs This paper investigates the gradient flow structure, well-posedness, and asymptotic behavior of the Fokker-Planck equation defined on locally uniformly finite graphs, which is highly non-trivial compared with the finite case. We first construct a 2-Wasserstein-type metric and gradient flow equation in the probability density space associated with the underlying graphs. Then, we prove the global existence of solution to the Fokker-Planck equation using a novel approach that differs significantly from the methods applied in the finite case. We also demonstrate that the solution converges to the Gibbs distribution in the $\ell^{r}(V,\bmπ)$ norm with $r\in [2,\infty]$, by using the indicator set partitioning method. To the best of our knowledge, this work seems the first result on the study of Wasserstein-type metrics and the Fokker-Planck equation in probability density spaces defined on infinite graphs. |
| title | Gradient flow structure, well-posedness and asymptotic behavior of Fokker-Planck equation on locally finite graphs |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2503.03531 |