Log-Sobolev Inequalities and Exponential Ergodicity for Non-degenerate and Degenerate McKean-Vlasov SDEs
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| Format: | Preprint |
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2025
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| _version_ | 1866911260041281536 |
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| author | Huang, Xing Kopfer, Eva Ren, Panpan |
| author_facet | Huang, Xing Kopfer, Eva Ren, Panpan |
| contents | The exponential ergodicity of partially dissipative McKean-Vlasov SDEs in the \(L^1\)-Wasserstein distance has been extensively studied using asymptotic reflection coupling. However, the reflection coupling method is not applicable for the exponential ergodicity in $L^2$-Wasserstein distance and relative entropy. In this paper, we first establish uniform log-Sobolev inequalities (in the frozen measure variable with bounded second moments) for the invariant probability measure of the corresponding SDEs with frozen distribution. Second, for the McKean-Vlasov SDEs, we combine the log-Harnack inequality and Talagrand's inequality to derive exponential ergodicity in both $L^2$-Wasserstein distance and relative entropy. Furthermore, we extend these main results to the case of degenerate diffusion. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_16092 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Log-Sobolev Inequalities and Exponential Ergodicity for Non-degenerate and Degenerate McKean-Vlasov SDEs Huang, Xing Kopfer, Eva Ren, Panpan Probability The exponential ergodicity of partially dissipative McKean-Vlasov SDEs in the \(L^1\)-Wasserstein distance has been extensively studied using asymptotic reflection coupling. However, the reflection coupling method is not applicable for the exponential ergodicity in $L^2$-Wasserstein distance and relative entropy. In this paper, we first establish uniform log-Sobolev inequalities (in the frozen measure variable with bounded second moments) for the invariant probability measure of the corresponding SDEs with frozen distribution. Second, for the McKean-Vlasov SDEs, we combine the log-Harnack inequality and Talagrand's inequality to derive exponential ergodicity in both $L^2$-Wasserstein distance and relative entropy. Furthermore, we extend these main results to the case of degenerate diffusion. |
| title | Log-Sobolev Inequalities and Exponential Ergodicity for Non-degenerate and Degenerate McKean-Vlasov SDEs |
| topic | Probability |
| url | https://arxiv.org/abs/2501.16092 |