[论文解读] Geometric properties of Dirichlet forms under order isomorphisms

Daniel Lenz, Marcel Schmidt+1 more|arXiv (Cornell University)|Jan 25, 2018

Geometric Analysis and Curvature Flows参考文献 32被引用 6

一句话总结

本文证明了在狄利克雷型式之间保持序关系的同构映射，会保持几何、拓扑和测度结构。在较弱假设下，此类同构映射分别在强局部与非局部正则狄利克雷型式中，诱导出内蕴度量与电阻度量下的等距同构，从而表明扩散过程决定了底层几何结构。

ABSTRACT

We study pairs of Dirichlet forms related by an intertwining order isomorphisms between the associated $L^2$-spaces. We consider the measurable, the topological and the geometric setting respectively. In the measurable setting, we deal with arbitrary (irreducible) Dirichlet forms and show that any intertwining order isomorphism is necessarily unitary (up to a constant). In the topological setting we deal with quasi-regular forms and show that any intertwining order isomorphism induces a quasi-homeomorphism between the underlying spaces. In the geometric setting we deal with both regular Dirichlet forms as well as resistance forms and essentially show that the geometry defined by these forms is preserved by intertwining order isomorphisms. In particular, we prove in the strongly local regular case that intertwining order isomorphisms induce isometries with respect to the intrinsic metrics between the underlying spaces under fairly mild assumptions. This applies to a wide variety of metric measure spaces including $\mathrm{RCD}(K,N)$-spaces, complete weighted Riemannian manifolds and complete quantum graphs. In the non-local regular case our results cover in particular graphs as well as fractional Laplacians as arising in the treatment of $α$-stable Lévy processes. For resistance forms we show that intertwining order isomorphisms are isometries with respect to the resistance metrics. Our results can can be understood as saying that diffusion always determines the Hilbert space, and -- under natural compatibility assumptions -- the topology and the geometry respectively. As special instances they cover earlier results for manifolds and graphs.

研究动机与目标

• 研究通过L²空间之间的序同构，狄利克雷型式是否能确定空间的几何结构。
• 确定在狄利克雷型式之间的交错序同构下，拓扑与几何结构在多大程度上被保持。
• 将已知关于流形与图的结果推广至一般狄利克雷空间，包括分形、加权黎曼流形与量子图。
• 建立在电阻型式上不可约且常返的狄利克雷型式，通过电阻度量等距同构在这些同构下保持不变。

提出的方法

• 利用Lamperti型定理，将序同构表示为可测变换与乘性函数的复合。
• 应用Beurling-Deny分解以分析狄利克雷型式及其相关半群的结构。
• 对强局部正则狄利克雷型式使用内蕴度量，并在较弱正则性条件下证明交错同构诱导出等距同构。
• 使用电阻度量公式 $ R(x,y) = \sup\{ |f(x)-f(y)|^2 : f \in \mathcal{F} \cap C_b(X), \mathcal{E}(f) \leq 1 \} $ 来刻画电阻型式。
• 证明同构所诱导的变换是同胚，且乘性因子几乎处处为常数。
• 应用交错性质 $ Q_2(Uf) = \|U\|^2 Q_1(f) $ 关联狄利克雷能量并推导出度量保持性。

实验结果

研究问题

• RQ1 两个狄利克雷型式的L²空间之间的序同构在多大程度上保持了底层几何结构？
• RQ2 在何种条件下，狄利克雷型式之间的交错序同构会在底层拓扑空间之间诱导出同胚？
• RQ3 狄利克雷型式的内蕴度量或电阻度量是否能在序同构下保持不变？若能，需满足何种条件？
• RQ4 在度量测度空间（如RCD(K,N)-空间或量子图）中，其几何结构是否能通过此类同构被其关联的狄利克雷型式完全确定？
• RQ5 变换映射τ与权函数h的性质如何与几何结构保持性相关联？

主要发现

• 任意不可约狄利克雷型式L²空间之间的交错序同构均为常数倍单位算子，保持测度结构。
• 在拓扑设定下，此类同构对拟正则狄利克雷型式在底层空间之间诱导出拟同胚。
• 对于强局部正则狄利克雷型式，在较弱假设下，同构诱导出内蕴度量下的等距同构。
• 对于电阻型式，同构诱导出电阻度量下的等距同构，且变换映射τ为同胚。
• 在紧致且有限测度的电阻型式情形下，同构在电阻度量空间之间诱导出满等距同构。
• 变换中的常数α满足α = ||U||，当参考测度为概率测度时，确保完全度量等距同构。