Papers - Atsuji, Atsushi

Nevanlinna Theory on Infinite Graphs

Atsuji A., Kaneko H.

Computational Methods and Function Theory  2024

ISSN  16179447

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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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A Chip-Firing and a Riemann-Roch Theorem on an Ultrametric Space

Atsuji A., Kaneko H.

Springer Proceedings in Mathematics and Statistics (Springer Proceedings in Mathematics and Statistics)  394   23 - 43 2022

ISSN  21941009

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A Riemann-Roch theorem on an edge-weighted infinite graph with local finiteness was established by the present authors in [1], where the spectral gap of Laplacian associated determined by the edge-weight was investigated as the corner stone of the proof. On the other hand, as for non-archimedean metric space, the Laplacians in the construction of Hunt processes such as in [3, 5] based on the Dirichlet space theory can be highlighted. However, in those studies, a positive edge-weight was given substantially between each pair of balls with an identical diameter with respect to the ultrametric and the spectral gap is infeasible. In the present article, we rethink the notion of chip-firing and show an upper bound of function given by accumulation of chip-firing to materialize a counterpart of the dimension of linear system in ultrametric space. In the final section of this article, we establish a Riemann-Roch theorem on an ultrametric space.

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Default functions and Liouville type theorems based on symmetric diffusions

Atsuji A.

Journal of the Mathematical Society of Japan (Journal of the Mathematical Society of Japan)  73 ( 2 ) 525 - 551 2021.04

ISSN  00255645

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Default functions appear when one discusses conditions which ensure that a local martingale is a true martingale. We show vanishing of default functions of Dirichlet processes enables us to obtain Liouville type theorems for subharmonic functions and holomorphic maps.

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A Riemann-Roch Theorem on Infinite Graphs

Atsushi Atsuji, Hiroshi Kaneko

Advances in Non-Archimedean Analysis and Applications (Springer)     297 - 312 2021

Research paper (international conference proceedings), Joint Work, Accepted

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Nevanlinna-Type theorems for meromorphic functions on non-positively curved Kähler manifolds

Atsuji A.

Forum Mathematicum (Forum Mathematicum)  30 ( 1 ) 171 - 189 2018

Research paper (scientific journal), Accepted,  ISSN  09337741

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© 2017 Walter de Gruyter GmbH, Berlin/Boston 2018. We give a second main theorem of Nevanlinna theory on complete non-positively curved Kähler manifolds. Its remainder term depends only on Ricci curvature of the manifolds except for the terms depending only on the characteristic functions.

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Value distribution of leafwise holomorphic maps on complex laminations by hyperbolic Riemann surfaces

ATSUJI ATSUSHI

J. Math. Soc. Japan 69 ( 2 ) 1 - 25 2017

Research paper (scientific journal), Single Work, Accepted

The submartingale property and Liouville type theorems

ATSUJI ATSUSHI

manuscripta mathematica (Springer)  154 ( 1 ) 129 - 146 2017

Research paper (scientific journal), Single Work, Accepted

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Leafwise Brownian Motions and Some Function Theoretic Properties of Laminations

ATSUJI ATSUSHI

Potential Analysis (Springer)  48 ( 1 ) 85 - 113 2017

Research paper (scientific journal), Single Work, Accepted,  ISSN  09262601

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© 2017, Springer Science+Business Media Dordrecht. We discuss the value distribution of Borel measurable functions which are subharmonic or meromorphic along leaves on laminations. They are called leafwise subharmonic functions or meromorphic functions respectively. We consider cases that each leaf is a negatively curved Riemannian manifold or Kähler manifold. We first consider the case when leaves are Riemannian with a harmonic measure in L.Garnett sense. We show some Liouville type theorem holds for leafwise subharmonic functions in this case. In the case of laminations whose leaves are Kähler manifolds with some curvature condition we consider the value distribution of leafwise meromorphic functions. If a lamination has an ergodic harmonic measure, a variant of defect relation in Nevanlinna theory is obtained for almost all leaves. It gives a bound of the number of omitted points by those functions. Consequently we have a Picard type theorem for leafwise meromorphic functions.

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ブラウン運動と複素解析

ATSUJI ATSUSHI

数理科学 (サイエンス社)   2014

幾何学的関数論と拡散過程が関連する話題から : 複素葉層構造を中心に

ATSUJI ATSUSHI

京都大学数理解析研究所講究録 (京都大学数理解析研究所)  1855   47 - 55 2013

Research paper (conference, symposium, etc.), Single Work

On the number of omitted values by a meromorphic function of finite energy and heat diffusions.

ATSUJI ATSUSHI

Journal of Geometric analysis 20 ( 4 ) 1008-1025 2010

Research paper (scientific journal), Single Work, Accepted

Estimates on the number of the omitted values by meromorphic functions. Probabilistic approach to geometry

ATSUJI ATSUSHI

Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo 57   49-59 2010

Research paper (conference, symposium, etc.), Single Work, Accepted

A second main theorem of Nevanlinna theory for meromorphic functions on complete Kähler manifolds

ATSUJI ATSUSHI

Jour. Math. Soc. Japan 60   471-490 2008

Research paper (scientific journal), Single Work, Accepted

A second main theorem of Nevanlinna theory for meromorphic functions on complex submanifolds in Cn

ATSUJI ATSUSHI

Potential analysis 29   119-138 2008

Research paper (scientific journal), Single Work, Accepted

Parabolicity, the divergence theorem for delta-subharmonic functions and applications to the Liouville theorems for harmonic maps.

ATSUJI ATSUSHI

The Tohoku Mathematical Journal 27 ( 3 ) 353-373 2005

Research paper (scientific journal), Single Work, Accepted

Parabolicity, Projective volume snd finiteness of total curvature of minimal surfaces

ATSUJI ATSUSHI

Kodai Mathematical Journal 27 ( 3 ) 227-236 2004

Research paper (scientific journal), Single Work, Accepted

ブラウン運動と調和写像・正則写像の値分布論.

ATSUJI ATSUSHI

数学 54 ( 3 ) 235-248 2002

Research paper (scientific journal), Single Work, Accepted

A lemma of logarithmic derivative for some delta-subharmonic functions.

ATSUJI ATSUSHI

Complex Variables. Theory and Application 46 ( 3 ) 195-206 2001

Research paper (scientific journal), Single Work, Accepted,  ISSN  0278-1077

Nevanlinna theory and stochastic calculus---Some applications of stochastic first main theorem.

ATSUJI ATSUSHI

Proceedings of the Second ISAAC Congress (International Society for Analysis, Applications and Computation)  1   427-432 2000

Research paper (conference, symposium, etc.), Single Work