Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory

By Rupert L. Frank, Daniel Lenz, D. Wingert. 2010

129 citations recorded by the source corpus.

We present a study of what may be called an intrinsic metric for a general regular Dirichlet form. For such forms we then prove a Rademacher type theorem. For strongly local forms we show existence of a maximal intrinsic metric (under a weak continuity condition) and for Dirichlet forms with an absolutely continuous jump kernel we characterize intrinsic metrics by bounds on certain integrals. We then turn to applications on spectral theory and provide for (measure perturbation of) general regular Dirichlet forms an Allegretto-Piepenbrinck type theorem, which is based on a ground state transform, and a Shnol type theorem. Our setting includes Laplacian on manifolds, on graphs and α-stable processes.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e908c06af8ecc3a56c026f6fdcc8319a96dec019

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Volume growth and stochastic completeness of graphs (2012, methodology): “The condition on the metric in the hypotheses of Theorem 1.2 appeared previously in work of the author in [10] and has also been used in papers of Frank, Lenz, and Wingert ([12]) and Grigor’yan, Huang, and Masamune ([17]).”
Ollivier Ricci curvature for general graph Laplacians: Heat equation, Laplacian comparison, non-explosion and diameter bounds (2017, methodology): “hich was proven in [Hsu89; Var83]. One tempting explanation for the discrepancy of the decay rate between manifolds and graphs is the choice of the metric. Frequently, intrinsic metrics introduced in [FLW14] are used to describe the geometry of graphs with unbounded vertex degrees and to give analog”
Stochastic completeness for graphs with curvature dimension conditions (2015, methodology): “In order to deal with unbounded Laplacians, we need the following intrinsic metrics on graphs introduced in [FLW14].”