Geometry and Analysis of Dirichlet forms

Yuan Zhou

(Beijing University of Aeronautics ans Astronatics)

Event: ERC Workshop on Geometric Analysis on sub-Riemannian and Metric Spaces

Date: Oct 11, 2011, time: 15:30

Place: Centro De Giorgi, Scuola Normale Superiore

Abstract. Let \(\mathcal{E}\) 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 \((1, 2)\)-Poincare inequality. We first discuss the (non-)coincidence of the intrinsic length structure and the gradient structure. Then when the Ricci curvature of \(X\) is bounded from below in the sense of Lott-Sturm-Villani, the following are equivalent:
(i) the heat flow of \(\mathcal{E}\) gives the unique gradient flow of the entropy \(\mathcal{U}_\infty\), |
(ii) \(\mathcal{E}\) 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 symptotics of the gradient of the heat kernel.