Note on basic features of large time behaviour of heat kernels

TL;DR

This paper investigates the long-term behavior of heat kernels associated with various Laplacian operators across different geometric and graph structures, revealing convergence properties and ground state energy relations.

Contribution

It provides a unified analysis of large time heat kernel behavior for positivity improving selfadjoint semigroups, encompassing manifolds, metric graphs, and discrete graphs.

Findings

Semigroup converges to the ground state over time

Averaged logarithms of kernels approach the ground state energy

Framework applies to Laplacians on diverse structures

Abstract

Large time behaviour of heat semigroups (and more generally, of positive selfadjoint semigroups) is studied. Convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoint semigroups. This framework includes Laplacians on manifolds, metric graphs and discrete graphs.