Let P_t be the (Neumann) diffusion semigroup P_t generated by a weighted Laplacian on a complete connected Riemannian manifold M without boundary or with a convex boundary. It is well known that the Bakry-Emery curvature is bounded below by a positive constant ≪> 0 if and only if$$W_{p}(\mu_{1}P_{t}, \mu_{2}P_{t})\leq e^{-\ll t} W_{p} (\mu_{1},\mu_{2}),\ \ t\geq 0, p\geq 1 $$holds for all probability measures μ₁ and μ₂ on M, where W_p is the L^p Wasserstein distance induced by the Riemannian distance. In this paper, we prove the exponential contraction$$W_{p}(\mu_{1}P_{t}, \mu_{2}P_{t})\leq ce^{-\ll t} W_{p} (\mu_{1},\mu_{2}),\ \ p \geq 1, t\geq 0$$for some constants c,≪> 0 for a class of diffusion semigroups with negative curvature where the constant c is essentially larger than 1. Similar results are derived for SDEs with multiplicative noise by using explicit conditions on the coefficients, which are new even for SDEs with additive noise.

中文翻译：

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负曲率扩散半群在Wasserstein距离中的指数收缩

让P_吨是穿孔（Neumann）扩散半群P_吨由上一个完整的连接的黎曼流形的加权拉普拉斯生成中号无边界或有凸边界。众所周知，当且仅当$$ W_ {p}（\ mu_ {1} P_ {t}，\ mu_ {2} P_ {t}时，Bakry-Emery曲率的下限为正常数≪> 0 ）\当量ë^ { - \ LL吨} W_ {p}（\ mu_ {1}，\ mu_ {2}），\ \吨\ GEQ 0，p \ GEQ 1 $$适用于所有概率测度μ ₁和μ ₂上中号，其中w ^ _p是大号^p黎曼距离引起的Wasserstein距离。在本文中，我们证明了指数收缩$$ W_ {p}（\ mu_ {1} P_ {t}，\ mu_ {2} P_ {t}）\ leq ce ^ {-\ ll t} W_ {p} （\ mu_ {1}，\ mu_ {2}），\ \ p \ geq 1，t \ geq 0 $$对于某些常数c，对于具有负曲率的一类扩散半群，≪> 0，其中常数c本质上是大于1。通过在系数上使用显式条件，对具有乘法噪声的SDE得出相似的结果，这对于具有加性噪声的SDE也是新的。