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

TL;DR

This paper develops a unified framework for intrinsic metrics associated with general regular Dirichlet forms, establishing key theorems and applications in spectral theory across various settings like manifolds, graphs, and stable processes.

Contribution

It introduces a general intrinsic metric concept for Dirichlet forms, proves a Rademacher theorem, and applies these to spectral theory including Allegretto-Piepenbrinck and Shnol type theorems.

Findings

Existence of a maximal intrinsic metric for strongly local forms.

Characterization of intrinsic metrics via integral bounds for jump kernels.

Spectral theory results including ground state transform and spectral bounds.

Abstract

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 $\alpha$-stable processes.