Geometry and analysis of Dirichlet forms. II. (English) Zbl 1297.53033
Summary: Given a regular, strongly local Dirichlet form \(\mathcal{E}\), under assumption that the lower bound of the Ricci curvature of Bakry-Emery, the local doubling and local Poincaré inequalities are satisfied, we obtain that: (i) the intrinsic differential and distance structures of \(\mathcal{E}\) coincide (ii) the Cheeger energy functional \(\operatorname{Ch}_{d_{\mathcal{E}}}\) is a quadratic norm. This shows that (ii) is necessary for the Riemannian Ricci curvature defined by Ambrosio-Gigli-Savaré to be bounded from below. This together with some recent results of Ambrosio-Gigli-Savaré yields that the heat flow gives a gradient flow of Boltzman-Shannon entropy under the above assumptions. We also obtain an improvement on Kuwada’s duality theorem for Dirichlet forms under the assumptions of doubling and Poincaré inequalities. Finally, Dirichlet forms are constructed to show that doubling and Poincaré inequalities are not enough to obtain either (i) or (ii) above; that is, the lower bound of the Bakry-Emery curvature condition is essential.
For Part I see [P. Koskela and Y. Zhou, Adv. Math. 231, No. 5, 2755–2801 (2012; Zbl 1253.53035)].
For Part I see [P. Koskela and Y. Zhou, Adv. Math. 231, No. 5, 2755–2801 (2012; Zbl 1253.53035)].
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 35K05 | Heat equation |
Keywords:
Dirichlet form; Cheeger energy functional; Bakry-Emery’s Ricci curvature; riemannian Ricci curvature; local Poincaré inequalities; Boltzman-Shannon entropyCitations:
Zbl 1253.53035References:
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