Note on basic features of large time behaviour of heat kernels

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 encompasses all irreducible semigroups coming from Dirichlet forms as well as suitable perturbations thereof. It includes, in particular, Laplacians on connected manifolds, metric graphs and discrete graphs.

Coverings and the heat equation on graphs: Stochastic incompleteness, the Feller property, and uniform transience

“…for any x, y ∈ X, see [38,57]. The result is then immediate since p t ( x, y) ≤ p t (x, y) which follows from Theorem 4.4.…”

“…By applying an analogue to a theorem of Li [61], proven in the graph setting in [38,57], we have the following result concerning λ 0 (L). Furthermore, we have equality if the covering has finitely many sheets.…”

“…Proof. It is always true that the limit above exists, see [38,57]. Now, if λ 0 ( L) = 0, then lim t→∞ p t ( x, y) = 0 by Corollary 8.2 in [38] as m( X) = ∞ since the number of sheets is infinite.…”

Sharp Davies–Gaffney–Grigor’yan Lemma on graphs

“…The sharpness of the term e −λt in (2) follows from the long time heat kernel behavior, i.e. the exponential decay related to the bottom of the ℓ 2 spectrum of Laplacian, which was first proved by Li [Li86] on Riemannian manifolds and extended to graphs with unbounded Laplacians by Keller et al [KLVW,Corollary 5.6].…”

Global properties of Dirichlet forms in terms of Green’s formula

“…As discussed in the general setting we will restrict our investigation to irreducible Dirichlet forms. We can characterize irreducibility in terms of connectedness of the underlying graph (see [19,17] as well). Lemma 6.9.…”