Dirichlet型的几何与分析
Geometry and Analysis of Dirichlet forms
摘要：
设$ \mathscr E $是$L^2(X, m)$上的正则强局部Dirichlet型，$d$是相应的本征距离.假设$d$诱导的拓扑与$ X$上的原拓扑一致，且$X$是紧的，满足一个加倍性质，支持弱$(1, 2)$ -Poincaré不等式.我们首先讨论了内禀长度结构和梯度结构的（非）重合性。在进一步假设$X$的里奇曲率在Lott-Sturm-Villani意义下是有界的条件下，证明了下列等价性：（i）$\mathscr E$的热流给出了$\mathscr U_\infty$的唯一梯度流，（ii）$\mathscr E$满足牛顿性质，（iii）内禀长度结构与梯度结构一致.此外，标准（电阻）Dirichlet形式的Sierpinski垫片配备Kusuoka措施，我们确定的内在长度结构与可测黎曼和梯度结构。我们还将上述结果应用于热核梯度的（粗）里奇曲率和渐近性。
英文摘要：
Let $ \mathscr E $ be a regular, strongly local Dirichlet form on $L^2(X, m)$ and $d$ the associated intrinsic distance. Assume that the topology induced by $d$ coincides with the original topology on $ X$, and that $X$ is compact, satisfies a doubling property and supports a weak $(1, 2)$-Poincaré inequality. We first discuss the (non-)coincidence of the intrinsic length structure and the gradient structure. Under the further assumption that the Ricci curvature of $X$ is bounded from below in the sense of Lott-Sturm-Villani, the following are shown to be equivalent: (i) the heat flow of $\mathscr E$ gives the unique gradient flow of $\mathscr U_\infty$, (ii) $\mathscr E$ satisfies the Newtonian property, (iii) the intrinsic length structure coincides with the gradient structure. Moreover, for the standard (resistance) Dirichlet form on the Sierpinski gasket equipped with the Kusuoka measure, we identify the intrinsic length structure with the measurable Riemannian and the gradient structures. We also apply the above results to the (coarse) Ricci curvatures and asymptotics of the gradient of the heat kernel.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
