Exponential Ergodicity for McKean-Vlasov SDEs with Singular Interactions
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915451657781248 |
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| author | Huang, Xing Wang, Feng-Yu |
| author_facet | Huang, Xing Wang, Feng-Yu |
| contents | Let $k\in (d,\infty]$ and consider the $k*$-distance $$\|μ-ν\|_{k*}:= \sup\Big\{|μ(f)-ν(f)|:\ f\in\B_b(\R^d),\ \|f\|_{\tt L^k}:=\sup_{x\in \R^d}\|1_{B(x,1)}f\|_{L^k}\le 1\Big\}$$ between probability measures on $\R^d$. The exponential ergodicity in $1$-Wasserstein and $k*$ distances is derived for a class of McKean-Vlasov SDEs with small singular interactions measured by $\|\cdot\|_{k*}.$ Moreover, the exponential ergodicity in $2$-Wasserstein distance and relative entropy is derived when the interaction term is given by
$$b^{(0)}(x,μ) :=\int_{\R^d}h(x-y)μ(\d y)$$ for some measurable function $h:\R^d\to\R^d$ with small $\|h\|_{\tt L^k}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13924 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exponential Ergodicity for McKean-Vlasov SDEs with Singular Interactions Huang, Xing Wang, Feng-Yu Probability Let $k\in (d,\infty]$ and consider the $k*$-distance $$\|μ-ν\|_{k*}:= \sup\Big\{|μ(f)-ν(f)|:\ f\in\B_b(\R^d),\ \|f\|_{\tt L^k}:=\sup_{x\in \R^d}\|1_{B(x,1)}f\|_{L^k}\le 1\Big\}$$ between probability measures on $\R^d$. The exponential ergodicity in $1$-Wasserstein and $k*$ distances is derived for a class of McKean-Vlasov SDEs with small singular interactions measured by $\|\cdot\|_{k*}.$ Moreover, the exponential ergodicity in $2$-Wasserstein distance and relative entropy is derived when the interaction term is given by $$b^{(0)}(x,μ) :=\int_{\R^d}h(x-y)μ(\d y)$$ for some measurable function $h:\R^d\to\R^d$ with small $\|h\|_{\tt L^k}$. |
| title | Exponential Ergodicity for McKean-Vlasov SDEs with Singular Interactions |
| topic | Probability |
| url | https://arxiv.org/abs/2508.13924 |