Geometry and analysis of Dirichlet forms (II)

Given a regular, strongly local Dirichlet form EE, 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 EE coincide;(ii) the Cheeger energy functional ChdEChdE 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.