The [Formula omitted]-Liouville Property on Graphs
In this paper we investigate the [Formula omitted]-Liouville property, underlining its connection with stochastic completeness and other structural features of the graph. We give a characterization of the [Formula omitted]-Liouville property in terms of the Green function of the graph and use it to...
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Published in | The Journal of fourier analysis and applications Vol. 29; no. 4 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Springer
01.08.2023
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Online Access | Get full text |
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Summary: | In this paper we investigate the [Formula omitted]-Liouville property, underlining its connection with stochastic completeness and other structural features of the graph. We give a characterization of the [Formula omitted]-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 [Formula omitted]-Liouville property and prove some comparison theorems for general graphs based on inner-outer curvatures. We also introduce the Dirichlet [Formula omitted]-Liouville property of subgraphs and prove that if a graph has a Dirichlet [Formula omitted]-Liouville subgraph, then it is [Formula omitted]-Liouville itself. As a consequence, we obtain that the [Formula omitted]-Liouville property is not affected by a finite perturbation of the graph and, just as in the continuous setting, a graph is [Formula omitted]-Liouville provided that at least one of its ends is Dirichlet [Formula omitted]-Liouville. |
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ISSN: | 1069-5869 1531-5851 |
DOI: | 10.1007/s00041-023-10025-3 |