6533b829fe1ef96bd128a401

RESEARCH PRODUCT

Geometry and analysis of Dirichlet forms (II)

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Abstract Given a regular, strongly local Dirichlet form E , under assumption that the lower bound of the Ricci curvature of Bakry–Emery, the local doubling and local Poincare inequalities are satisfied, we obtain that: (i) the intrinsic differential and distance structures of E coincide; (ii) the Cheeger energy functional Ch d E is a quadratic norm. This shows that (ii) is necessary for the Riemannian Ricci curvature defined by Ambrosio–Gigli–Savare to be bounded from below. This together with some recent results of Ambrosio–Gigli–Savare 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 Poincare inequalities. Finally, Dirichlet forms are constructed to show that doubling and Poincare inequalities are not enough to obtain either (i) or (ii) above; that is, the lower bound of the Bakry–Emery curvature condition is essential.