Sharp $L^q$-Convergence Rate in $p$-Wasserstein Distance for Empirical Measures of Diffusion Processes
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866911993106006016 |
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| author | Wang, Feng-Yu Wu, Bingyao Zhu, Jie-Xiang |
| author_facet | Wang, Feng-Yu Wu, Bingyao Zhu, Jie-Xiang |
| contents | For a class of (non-symmetric) diffusion processes on a length space, which in particular include the (reflecting) diffusion processes on a connected compact Riemannian manifold, the exact convergence rate is derived for $({\mathbb E} [{\mathbb W}_p^q(μ_T,μ)])^{\frac{1}{q}} (T \to \infty)$ uniformly in $(p,q)\in [1,\infty) \times (0,\infty)$, where $μ_T$ is the empirical measure of the diffusion process, $μ$ is the unique invariant probability measure, and ${\mathbb W}_p$ is the $p$-Wasserstein distance. Moreover, when the dimension parameter is less than $4$, we prove that ${\mathbb E} |T {\mathbb W}_2^2(μ_T,μ)-Ξ(T)|^q \to 0$ as $T\to\infty$ for any $q\ge 1$, where $Ξ(T)$ is explicitly given by eigenvalues and eigenfunctions for the symmetric part of the generator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09116 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp $L^q$-Convergence Rate in $p$-Wasserstein Distance for Empirical Measures of Diffusion Processes Wang, Feng-Yu Wu, Bingyao Zhu, Jie-Xiang Probability For a class of (non-symmetric) diffusion processes on a length space, which in particular include the (reflecting) diffusion processes on a connected compact Riemannian manifold, the exact convergence rate is derived for $({\mathbb E} [{\mathbb W}_p^q(μ_T,μ)])^{\frac{1}{q}} (T \to \infty)$ uniformly in $(p,q)\in [1,\infty) \times (0,\infty)$, where $μ_T$ is the empirical measure of the diffusion process, $μ$ is the unique invariant probability measure, and ${\mathbb W}_p$ is the $p$-Wasserstein distance. Moreover, when the dimension parameter is less than $4$, we prove that ${\mathbb E} |T {\mathbb W}_2^2(μ_T,μ)-Ξ(T)|^q \to 0$ as $T\to\infty$ for any $q\ge 1$, where $Ξ(T)$ is explicitly given by eigenvalues and eigenfunctions for the symmetric part of the generator. |
| title | Sharp $L^q$-Convergence Rate in $p$-Wasserstein Distance for Empirical Measures of Diffusion Processes |
| topic | Probability |
| url | https://arxiv.org/abs/2408.09116 |