2023/05/02 by Adriani, Andrea, Setti, Alberto G.
#Differential Geometry (math.DG) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2305.01300
In this paper we investigate the L1 -Liouville property, underlining its connection with stochastic completeness and other structural features of the graph. We give a characterization of the L1 -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 L1 -Liouville property and prove some comparison theorems for general graphs based on inner-outer curvatures. We also introduce the Dirichlet L1-Liouville property of subgraphs and prove that if a graph has a Dirichlet L1-Liouville subgraph, then it is L1-Liouville itself. As a consequence, we obtain that the L1-Liouville property is not affected by a finite perturbation of the graph and, just as in the continuous setting, a graph is L1-Liouville provided that at least one of its ends is Dirichlet L1-Liouville.