McKean-Vlasov SDEs with Local Distributional Interactions: Well-Posedness and Entropy-Cost Estimates
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| Format: | Preprint |
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2026
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| author | Huang, Xing Ren, Panpan Wang, Feng-Yu |
| author_facet | Huang, Xing Ren, Panpan Wang, Feng-Yu |
| contents | We study McKean-Vlasov SDEs with interaction kernels in $\tt W^{-\dd,k},$ the local negative Sobolev space on $\R^d$ with indexes $\dd \in [0,\infty)$ and $k\in [1,\infty].$ We derive the local well-posedness for any singular indexes $(\dd,k)\in [0,\infty)\times [1,\infty],$ and prove the global well-posedness
for any initial distributions provided $\dd+\ff d k<1$. Moreover, the relative entropy and the $\|\cdot\|_{\dd,k*}$-distance induced by $ \tt W^{-\dd,k}$ are estimated for the time-marginal distributions of solutions by using the Wasserstein distance of initial distributions, which describe the regularity of the solution in initial distribution. In particular, the main results apply to Nemytskii-type SDEs which depend on higher order derivatives of the density functions, as well as McKean-Vlasov SDEs with interactions more singular than Riesz kernels. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_10841 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | McKean-Vlasov SDEs with Local Distributional Interactions: Well-Posedness and Entropy-Cost Estimates Huang, Xing Ren, Panpan Wang, Feng-Yu Probability We study McKean-Vlasov SDEs with interaction kernels in $\tt W^{-\dd,k},$ the local negative Sobolev space on $\R^d$ with indexes $\dd \in [0,\infty)$ and $k\in [1,\infty].$ We derive the local well-posedness for any singular indexes $(\dd,k)\in [0,\infty)\times [1,\infty],$ and prove the global well-posedness for any initial distributions provided $\dd+\ff d k<1$. Moreover, the relative entropy and the $\|\cdot\|_{\dd,k*}$-distance induced by $ \tt W^{-\dd,k}$ are estimated for the time-marginal distributions of solutions by using the Wasserstein distance of initial distributions, which describe the regularity of the solution in initial distribution. In particular, the main results apply to Nemytskii-type SDEs which depend on higher order derivatives of the density functions, as well as McKean-Vlasov SDEs with interactions more singular than Riesz kernels. |
| title | McKean-Vlasov SDEs with Local Distributional Interactions: Well-Posedness and Entropy-Cost Estimates |
| topic | Probability |
| url | https://arxiv.org/abs/2602.10841 |