Spectral bounds for exit times on metric measure Dirichlet spaces and applications

TL;DR

This paper establishes sharp spectral bounds for exit times of diffusions on metric measure spaces with sub-Gaussian heat kernel bounds, linking spectral properties to geometric and probabilistic behaviors.

Contribution

It provides a novel asymptotically sharp spectral upper bound on survival probabilities under sub-Gaussian heat kernel assumptions, with applications to geometry and fractals.

Findings

Finite mean exit time iff bottom of spectrum is positive

Spectral bounds imply bounds on Hot Spots constant

Results extend to sub-Riemannian manifolds and fractals

Abstract

Assuming the heat kernel on a doubling Dirichlet metric measure space has a sub-Gaussian bound, we prove an asymptotically sharp spectral upper bound on the survival probability of the associated diffusion process. As a consequence, we can show that the supremum of the mean exit time over all starting points is finite if and only if the bottom of the spectrum is positive. Among several applications, we show that the spectral upper bound on the survival probability implies a bound for the Hot Spots constant for Riemannian manifolds. Our results apply to interesting geometric settings including sub-Riemannian manifolds and fractals.