Geometry and analysis of Dirichlet forms

Let EE be a regular, strongly local Dirichlet form on L2(X,m)L2(X,m) and dd the associated intrinsic distance. Assume that the topology induced by dd coincides with the original topology on XX, and that XX is compact, satisfies a doubling property and supports a weak (1,2)(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 XX is bounded from below in the sense of Lott–Sturm–Villani, the following are shown to be equivalent: (i)the heat flow of EE gives the unique gradient flow of U∞U∞,(ii)EE 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.