论文标题
奇异空间上的配置空间-II。曲率
Configuration Spaces over Singular Spaces -- II. Curvature
论文作者
论文摘要
这是单个空间上的$ x $上的关于配置空间$υ$的系列的第二篇论文。在这里，我们专注于扩展度量度量空间$（υ，\ Mathsf {d}_υ，μ）$配备$ l^2 $ -Transportation距离$ \ Mathsf {D}_υ$和混合泊松量$μ$ $ $ $的几何方面。首先，我们建立了基本的自我相关性和从$ x $上提起的$υ$上的laplacian的$ l^p $ - 唯一性。其次，我们证明了Bakry-émery曲率在$ x $和$购$上的等效性，而$ x $没有任何公制假设。我们进一步证明了$υ$上的演变变化不平等，并引入了扩展度量度量空间$υ$的合成ricci-curvator下限的概念。作为一个应用程序，我们证明了Sobolev-to-lipschitz属性上的$υ$上的单数空间$ x $，最初是在$ x $的情况下猜想的。作为进一步的应用程序，我们证明了$ l^\ infty $ -to - $ \ MATHSF {d}_υ$ -lipschitz在$υ$上对热的半群的正则化，并以最佳运输方式对相应粒子系统的细微性进行了新的表征。
This is the second paper of a series on configuration spaces $Υ$ over singular spaces $X$. Here, we focus on geometric aspects of the extended metric measure space $(Υ, \mathsf{d}_Υ, μ)$ equipped with the $L^2$-transportation distance $\mathsf{d}_Υ$, and a mixed Poisson measure $μ$. Firstly, we establish the essential self-adjointness and the $L^p$-uniqueness for the Laplacian on $Υ$ lifted from $X$. Secondly, we prove the equivalence of Bakry-Émery curvature bounds on $X$ and on $Υ$, without any metric assumption on $X$. We further prove the Evolution Variation Inequality on $Υ$, and introduce the notion of synthetic Ricci-curvature lower bounds for the extended metric measure space $Υ$. As an application, we prove the Sobolev-to-Lipschitz property on $Υ$ over singular spaces $X$, originally conjectured in the case when $X$ is a manifold by M. Röckner and A. Schield. As a further application, we prove the $L^\infty$-to-$\mathsf{d}_Υ$-Lipschitz regularization of the heat semigroup on $Υ$ and gives a new characterization of the ergodicity of the corresponding particle systems in terms of optimal transport.
