Published papers

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.

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.

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.

© 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.

© 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.