In this paper we investigate the L-1-Liouville property, underlining its connection with stochastic completeness and other structural features of the graph. We give a characterization of the L-1-Liouville property in terms of the Green function of the graph and use it to prove its equivalence with stochastic completeness on model graphs. Moreover, we show that there exist stochastically incomplete graphs which satisfy the L-1-Liouville property and prove some comparison theorems for general graphs based on inner-outer curvatures. We also introduce the Dirichlet L-1-Liouville property of subgraphs and prove that if a graph has a Dirichlet L-1-Liouville subgraph, then it is L-1-Liouville itself. As a consequence, we obtain that the L-1-Liouville property is not affected by a finite perturbation of the graph and, just as in the continuous setting, a graph is L-1-Liouville provided that at least one of its ends is Dirichlet L-1-Liouville.

### The L1-Liouville Property on Graphs

#### Abstract

In this paper we investigate the L-1-Liouville property, underlining its connection with stochastic completeness and other structural features of the graph. We give a characterization of the L-1-Liouville property in terms of the Green function of the graph and use it to prove its equivalence with stochastic completeness on model graphs. Moreover, we show that there exist stochastically incomplete graphs which satisfy the L-1-Liouville property and prove some comparison theorems for general graphs based on inner-outer curvatures. We also introduce the Dirichlet L-1-Liouville property of subgraphs and prove that if a graph has a Dirichlet L-1-Liouville subgraph, then it is L-1-Liouville itself. As a consequence, we obtain that the L-1-Liouville property is not affected by a finite perturbation of the graph and, just as in the continuous setting, a graph is L-1-Liouville provided that at least one of its ends is Dirichlet L-1-Liouville.
##### Scheda breve Scheda completa Scheda completa (DC)
2023
2023
Infinite weighted graphs; L-1-Liouville property; Stochastic completeness; Curvature comparison
Adriani, A; Setti, Ag
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Adriani, Andrea; Setti, Alberto G. The L1-Liouville property on graphs. J. Fourier Anal. Appl. 29 (2023).pdf

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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11383/2160951`