Exponential Convergence in Entropy and Wasserstein Distance for McKean-Vlasov SDEs
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| Format: | Preprint |
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2020
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| _version_ | 1866910623736004608 |
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| author | Ren, Panpan Wang, Feng-Yu |
| author_facet | Ren, Panpan Wang, Feng-Yu |
| contents | The following type exponential convergence is proved for (non-degenerate or degenerate) McKean-Vlasov SDEs: $$W_2(μ_t,μ_\infty)^2 +{\rm Ent}(μ_t|μ_\infty)\le c {\rm e}^{-λt} \min\big\{W_2(μ_0, μ_\infty)^2,{\rm Ent}(μ_0|μ_\infty)\big\},\ \ t\ge 1,$$ where $c,λ>0$ are constants, $μ_t$ is the distribution of the solution at time $t$, $μ_\infty$ is the unique invariant probability measure, ${\rm Ent}$ is the relative entropy and $W_2$ is the $L^2$-Wasserstein distance. In particular, this type exponential convergence holds for some (non-degenerate or degenerate) granular media type equations generalizing those studied in [CMV, GLW] on the exponential convergence in a mean field entropy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2010_08950 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Exponential Convergence in Entropy and Wasserstein Distance for McKean-Vlasov SDEs Ren, Panpan Wang, Feng-Yu Probability The following type exponential convergence is proved for (non-degenerate or degenerate) McKean-Vlasov SDEs: $$W_2(μ_t,μ_\infty)^2 +{\rm Ent}(μ_t|μ_\infty)\le c {\rm e}^{-λt} \min\big\{W_2(μ_0, μ_\infty)^2,{\rm Ent}(μ_0|μ_\infty)\big\},\ \ t\ge 1,$$ where $c,λ>0$ are constants, $μ_t$ is the distribution of the solution at time $t$, $μ_\infty$ is the unique invariant probability measure, ${\rm Ent}$ is the relative entropy and $W_2$ is the $L^2$-Wasserstein distance. In particular, this type exponential convergence holds for some (non-degenerate or degenerate) granular media type equations generalizing those studied in [CMV, GLW] on the exponential convergence in a mean field entropy. |
| title | Exponential Convergence in Entropy and Wasserstein Distance for McKean-Vlasov SDEs |
| topic | Probability |
| url | https://arxiv.org/abs/2010.08950 |