Nevanlinna Theory and Stochastic Calculus

Abstract

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.