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.
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Acknowledgements
The authors thank the referee for careful reading of the manuscript and several helpful suggestions. This work was supported by JSPS KAKENHI Grant Numbers JP21K03299, JP21K03277.
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Communicated by Risto Korhonen.
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Atsuji, A., Kaneko, H. Nevanlinna Theory on Infinite Graphs. Comput. Methods Funct. Theory (2024). https://doi.org/10.1007/s40315-024-00530-x
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DOI: https://doi.org/10.1007/s40315-024-00530-x