Proceedings of the Second ISAAC Congress pp 427-432 | Cite as

# 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.

### Keywords

Manifold## Preview

Unable to display preview. Download preview PDF.

### References

- [1]A. Atsuji, “Nevanlinna theory via stochastic calculus.” Jour. Fanct. Anal. vol. 132, No. 2 1995, 437–510MathSciNetGoogle Scholar
- [2]A. Atsuji “A Casorati-Weierstrass theorem for holomorphic maps and invariant
*σ*—fields of holomorphic diffusions.” Bull.Sci.math. 123 (1999), 371–383.MathSciNetMATHCrossRefGoogle Scholar - [3]A. Atsuji, “Liouville property and minimal surfaces.” in preparation.Google Scholar
- [4]A. Atsuji, “Remarks on harmonic maps into a cone from stochastically complete manifolds.” Proc. Japan Acad. Ser. A 75 (1999), 105–108.MathSciNetMATHCrossRefGoogle Scholar
- [5]R. F. Bass,
*Probabilistic techniques in analysis*. Probability and its Applications. Springer-Verlag, New York, 1995.MATHGoogle Scholar - [6]T. K. Carne, “Brownian motion and Nevanlinna theory.” Jour Proc. London Math. Soc. vol 52, 1986, 349–368MathSciNetMATHCrossRefGoogle Scholar
- [7]R. W. R. Darling, “Convergence of martingales on a Riemannian manifold.” Publ.Res.Inst.Math.Kyoto Univ., 19 (1983), 753–763.MathSciNetMATHCrossRefGoogle Scholar
- [8]B. Davis, “Picard’s theorem and Brownian motion.” Trans. Amer. Math. Soc. 213 (1975), 353–362.MathSciNetMATHGoogle Scholar
- [9]B. Davis, “Brownian motion and analytic functions.” Ann. Probab. 7 (1979), no. 6, 913–932.MathSciNetMATHCrossRefGoogle Scholar
- [10][10] K. D. Elworthy, X. M. Li and M. Yor, “On the tails of the supremum and the quadratic variation of strictly local martingales.” Séminaire de Probabilités XXXI, Lect.Note Math.1655(1997), Springer.Google Scholar
- [11]R. E. Molzon, “Potential theory on complex projective space: application to characterization of pluripolar sets and growth of analytic varieties.” Ill.Jour.Math., 28 (1984), 103–119.MathSciNetMATHGoogle Scholar
- [12]H. Omori, “Isometric immersions of Riemannian manifolds.” J. Math. Soc. Japan, 19 (1967), 205–214.MathSciNetMATHCrossRefGoogle Scholar
- [13]S. T. Yau, “Harmonic functions on complete Riemannian manifolds.” Comm.Pure.Appl.Math., 28 (1975), 201–228.MathSciNetMATHCrossRefGoogle Scholar