Geometry and Analysis of Dirichlet forms

By Pekka Koskela, Yuan Zhou. 2012

72 citations recorded by the source corpus.

Let $ \mathscr 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 $(1, 2)$-Poincar\'e 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 $X$ is bounded from below in the sense of Lott-Sturm-Villani, the following are shown to be equivalent: (i) the heat flow of $\mathscr E$ gives the unique gradient flow of $\mathscr U_\infty$, (ii) $\mathscr E$ 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 asymptotics of the gradient of the heat kernel.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/903ded52024f898ee577731c9b95cc3e46b0b94e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces (2013, background): “The relation between the two points of view and the compatibility of metric measure structures and Energy structures have been discussed extensively in [5] as well as in [27].”
Cones over metric measure spaces and the maximal diameter theorem (2013, background): “The following theorem of Koskela and Zhou in [33] will be important later.”
Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates (2018, background): “In the context of fractals, strictly local Dirichlet forms appear in [4, 23, 24, 34, 47, 54,55,58,61,68,76, 91,92] and play an important role in analysis of first-order derivatives in these settings.”
The Splitting Theorem in Non-Smooth Context (2013, unknown): “There are indeed strong connections between the two points of view, in particular in relation with lower bounds on the Ricci curvature, see [7], [8] and [43] for recent progresses in this direction.”
Gradient estimates for heat kernels and harmonic functions (2017, background): “Under the requirements of doubling and Poincaré inequality (P2), it is known that W(1,2)(X) = N(1,2)(X) = W(1,2)(X), and that Lipschitz functions are dense in these spaces; see [99, 79].”
Space of Ricci Flows I (2009, background): “A good reference is the nice paper [67], by P.”