Graphs of finite measure

Abstract

We consider weighted graphs with an infinite set of vertices. We show that boundedness of all functions of finite energy can be seen as a notion of 'relative compactness' for such graphs and study sufficient and necessary conditions for this property in terms of various metrics. We then equip graphs satisfying this property with a finite measure and investigate the associated Laplacian and its semigroup. In this context, our results include the trace class property for the semigroup, uniqueness and existence of solutions to the Dirichlet problem with boundary arising from the natural compactification, an explicit description of the domain of the Dirichlet Laplacian, convergence of the heat semigroup for large times as well as stochastic incompleteness and transience of the corresponding random walk in continuous time.

Volume growth and bounds for the essential spectrum for Dirichlet forms

“…(d) If we assume that ρ takes values in [0, ∞), then we can clearly replace the assumption that m( r B r (x 0 )) = ∞ with m(X) = ∞. The case when m(X) < ∞ is notably different; see [8] for more details.…”

Diffusion determines the recurrent graph

“…The forms Q (D) and Q (N ) are of particular relevance in the theory of forms associated to graphs in the sense that a symmetric, closed quadratic form Q is associated to the graph if and only if Q (D) ⊆ Q ⊆ Q (N ) , see [19]. Here, we write…”

A Feynman–Kac–Itô formula for magnetic Schrödinger operators on graphs

“…Indeed, the only assumption in [28, Definition 2.3.1] which is non-trivial to check is (RF04). This however follows by [11,Lemma 3.4]. On the other hand, a magnetic form Q (c) 0,θ , θ = 0, can not be extended to a resistance form since it violates the cut-off property (RF05).…”