{"title":"Nevanlinna Theory on Infinite Graphs","authors":"Atsushi Atsuji, Hiroshi Kaneko","doi":"10.1007/s40315-024-00530-x","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we explore a generalization of one-dimensional tropical Nevanlinna theory developed by Halburd & Southall and Laine & Toghe for a scheme on general locally finite graphs. We first give a probabilistic interpretation of a fundamental observation in one-dimensional tropical Nevanlinna theory on the graph with countably infinitely many vertices of degree two, aiming at its extension in terms of one-dimensional Brownian motion. A counterpart of Lemma on the logarithmic derivative in the classical Nevanlinna theory was proved by Halburd and Southall (cf. Int. Math. Res. Not. 2009:887–911, 2009, https://doi.org/10.1093/imrn/rnn150). Taking advantage of the stochastic analytical interpretation, we prove an analogous result to their lemma on the logarithmic derivative on infinite graphs admitting tree structure.</p>","PeriodicalId":49088,"journal":{"name":"Computational Methods and Function Theory","volume":"152 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Methods and Function Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40315-024-00530-x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we explore a generalization of one-dimensional tropical Nevanlinna theory developed by Halburd & Southall and Laine & Toghe for a scheme on general locally finite graphs. We first give a probabilistic interpretation of a fundamental observation in one-dimensional tropical Nevanlinna theory on the graph with countably infinitely many vertices of degree two, aiming at its extension in terms of one-dimensional Brownian motion. A counterpart of Lemma on the logarithmic derivative in the classical Nevanlinna theory was proved by Halburd and Southall (cf. Int. Math. Res. Not. 2009:887–911, 2009, https://doi.org/10.1093/imrn/rnn150). Taking advantage of the stochastic analytical interpretation, we prove an analogous result to their lemma on the logarithmic derivative on infinite graphs admitting tree structure.
期刊介绍:
CMFT is an international mathematics journal which publishes carefully selected original research papers in complex analysis (in a broad sense), and on applications or computational methods related to complex analysis. Survey articles of high standard and current interest can be considered for publication as well.