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

2024/09/11 by Carfagnini, Marco, Gordina, Maria, Teplyaev, Alexander · 1 citation
#60G18 28A80 31C25 43A85 60F15 60F17 60G51 60J60 60J65 #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2409.07425

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.

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