Variational structure of Fokker-Planck equations with variable mobility
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908366730690560 |
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| author | Liu, Hailiang Tzavaras, Athanasios E. |
| author_facet | Liu, Hailiang Tzavaras, Athanasios E. |
| contents | We study Fokker--Planck equations with symmetric, positive definite mobility matrices capturing diffusion in heterogeneous environments. A weighted Wasserstein metric is introduced for which these equations are gradient flows. This metric is shown to emerge from an optimal control problem in the space of probability densities for a class of variable mobility matrices, with the cost function capturing the work dissipated via friction. Using the Nash-Kuiper isometric embedding theorem for Riemannian manifolds, we demonstrate the existence of optimal transport maps. Additionally, we construct a time-discrete variational scheme, establish key properties for the associated minimizing problem, and prove convergence to weak solutions of the associated Fokker-Planck equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_10676 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Variational structure of Fokker-Planck equations with variable mobility Liu, Hailiang Tzavaras, Athanasios E. Optimization and Control Analysis of PDEs 35A15, 35K55, 60J60 We study Fokker--Planck equations with symmetric, positive definite mobility matrices capturing diffusion in heterogeneous environments. A weighted Wasserstein metric is introduced for which these equations are gradient flows. This metric is shown to emerge from an optimal control problem in the space of probability densities for a class of variable mobility matrices, with the cost function capturing the work dissipated via friction. Using the Nash-Kuiper isometric embedding theorem for Riemannian manifolds, we demonstrate the existence of optimal transport maps. Additionally, we construct a time-discrete variational scheme, establish key properties for the associated minimizing problem, and prove convergence to weak solutions of the associated Fokker-Planck equation. |
| title | Variational structure of Fokker-Planck equations with variable mobility |
| topic | Optimization and Control Analysis of PDEs 35A15, 35K55, 60J60 |
| url | https://arxiv.org/abs/2505.10676 |