Wasserstein geometry of nonnegative measures on finite Markov chains I: Gradient flow
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912832237338624 |
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| author | Mao, Qifan Wang, Xinyu Xue, Xiaoping |
| author_facet | Mao, Qifan Wang, Xinyu Xue, Xiaoping |
| contents | We investigate a Benamou--Brenier type transportation metric for nonnegative measures on a finite reversible Markov chain, which endows the space of measures with a Riemannian structure. Using this geometric framework, we identify a generalized heat equation with source as the gradient flow of the discrete entropy. Moreover, by means of a local Łojasiewicz inequality, we prove exponential convergence of the flow to a unique equilibrium. Our results clarify the role of the Benamou--Brenier formulation in discrete optimal transport for nonnegative measures and provide a coherent geometric interpretation of generalized diffusion equations with source terms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_13073 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Wasserstein geometry of nonnegative measures on finite Markov chains I: Gradient flow Mao, Qifan Wang, Xinyu Xue, Xiaoping Analysis of PDEs Probability 49Q22, 60J27, 35K05 We investigate a Benamou--Brenier type transportation metric for nonnegative measures on a finite reversible Markov chain, which endows the space of measures with a Riemannian structure. Using this geometric framework, we identify a generalized heat equation with source as the gradient flow of the discrete entropy. Moreover, by means of a local Łojasiewicz inequality, we prove exponential convergence of the flow to a unique equilibrium. Our results clarify the role of the Benamou--Brenier formulation in discrete optimal transport for nonnegative measures and provide a coherent geometric interpretation of generalized diffusion equations with source terms. |
| title | Wasserstein geometry of nonnegative measures on finite Markov chains I: Gradient flow |
| topic | Analysis of PDEs Probability 49Q22, 60J27, 35K05 |
| url | https://arxiv.org/abs/2601.13073 |