Nevanlinna Theory and Stochastic Calculus

Some Applications of Stochastic First Main Theorem
  • Atsushi Atsuji
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 7)


Several relationships between classical function theory and analysis of Brownian motion have been studied by many authors (cf. [9], [5]). Davis gave an elegant proof of Picard’s theorem in [8] which impressed us with their intimacy. It also gave some possibility to study value-distribution of meromorphic functions through Brownian motion or stochastic calculus. It is curious that there had been no article noting a simple relation between Nevanlinna theory and Brownian motion until Carne’s paper [6]. He considered some probabilistic aspects of classical Nevanlinna theory. After the work the author improved it to several dimensional cases in [1]. In this note we exemplify this natural relation between stochastic calculus and Nevanlinna theory by considering a stochastic generalization of first main theorem of Nevanlinna and its applications.


Brownian Motion Algebraic Variety Stochastic Calculus Local Martingale Minimal Immersion 
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Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Atsushi Atsuji
    • 1
  1. 1.Department of Mathematics, Graduate School of ScienceOsaka UniversityToyonaka, OsakaJapan

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