Dirichlet metric measure spaces: spectrum, irreducibility, and small deviations

TL;DR

This paper investigates the spectral properties and probabilistic behaviors of ultracontractive irreducible Dirichlet metric measure spaces, establishing discreteness of the spectrum and analyzing small deviations and heat content asymptotics.

Contribution

It proves the discreteness of the Dirichlet spectrum on connected open sets without boundary regularity assumptions, extending spectral theory to broad classes of spaces.

Findings

Dirichlet spectrum is discrete on connected open sets

Established small deviation estimates for associated Hunt processes

Derived large time asymptotics for generalized heat content

Abstract

We show that for ultracontractive irreducible Dirichlet metric measure spaces, the Dirichlet spectrum is discrete for a restriction to any connected open set without any assumption on regularity of the boundary. The main applications include small deviations for the corresponding Hunt process and large time asymptotics for the generalized heat content. Our examples include Riemannian and sub-Riemannian manifolds, as well as non-smooth and fractal spaces.