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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.00314 |
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| _version_ | 1866914271190843392 |
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| author | Li, Ruowei Münch, Florentin |
| author_facet | Li, Ruowei Münch, Florentin |
| contents | In this paper, we prove the convergence and uniqueness of a general discrete-time nonlinear Markov chain with specific conditions. The results have important applications in discrete differential geometry. First, we prove the discrete-time Ollivier Ricci curvature flow $d_{n+1}:=(1-ακ_{d_{n}})d_{n}$ converges to a constant curvature metric on a finite weighted graph. As shown in \cite[Theorem 5.1]{M23}, a Laplacian separation principle holds on a locally finite graph with nonnegative Ollivier curvature. We further prove that the Laplacian separation flow converges to the constant Laplacian solution and generalize the result to nonlinear $p$-Laplace operators. Moreover, our results can also be applied to study the long-time behavior in the nonlinear Dirichlet forms theory and nonlinear Perron-Frobenius theory. Finally, we define the Ollivier Ricci curvature of the nonlinear Markov chain which is consistent with the classical Ollivier Ricci curvature, sectional curvature \cite{CMS24}, coarse Ricci curvature on hypergraphs \cite{IKTU21} and the modified Ollivier Ricci curvature for $p$-Laplace. We also establish the convergence results for the nonlinear Markov chain with nonnegative Ollivier Ricci curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_00314 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The convergence and uniqueness of a discrete-time nonlinear Markov chain Li, Ruowei Münch, Florentin Dynamical Systems 60J10, 60J20, 47D07, 53A70, 37C25 In this paper, we prove the convergence and uniqueness of a general discrete-time nonlinear Markov chain with specific conditions. The results have important applications in discrete differential geometry. First, we prove the discrete-time Ollivier Ricci curvature flow $d_{n+1}:=(1-ακ_{d_{n}})d_{n}$ converges to a constant curvature metric on a finite weighted graph. As shown in \cite[Theorem 5.1]{M23}, a Laplacian separation principle holds on a locally finite graph with nonnegative Ollivier curvature. We further prove that the Laplacian separation flow converges to the constant Laplacian solution and generalize the result to nonlinear $p$-Laplace operators. Moreover, our results can also be applied to study the long-time behavior in the nonlinear Dirichlet forms theory and nonlinear Perron-Frobenius theory. Finally, we define the Ollivier Ricci curvature of the nonlinear Markov chain which is consistent with the classical Ollivier Ricci curvature, sectional curvature \cite{CMS24}, coarse Ricci curvature on hypergraphs \cite{IKTU21} and the modified Ollivier Ricci curvature for $p$-Laplace. We also establish the convergence results for the nonlinear Markov chain with nonnegative Ollivier Ricci curvature. |
| title | The convergence and uniqueness of a discrete-time nonlinear Markov chain |
| topic | Dynamical Systems 60J10, 60J20, 47D07, 53A70, 37C25 |
| url | https://arxiv.org/abs/2407.00314 |