Electronic Communications in Probability

A martingale on the zero-set of a holomorphic function

Peter Kink

Full-text: Open access

Abstract

We give a simple probabilistic proof of the classical fact from complex analysis that the zeros of a holomorphic function of several variables are never isolated and that they are not contained in any compact set. No facts from complex analysis are assumed other than the Cauchy-Riemann definition. From stochastic analysis only the Ito formula and the standard existence theorem for stochastic differential equations are required.

Article information

Source
Electron. Commun. Probab., Volume 13 (2008), paper no. 55, 606-613.

Dates
Accepted: 24 November 2008
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465233482

Digital Object Identifier
doi:10.1214/ECP.v13-1425

Mathematical Reviews number (MathSciNet)
MR2461534

Zentralblatt MATH identifier
1189.60091

Subjects
Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60G46: Martingales and classical analysis 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
complex martingales stochastic differential equations

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Kink, Peter. A martingale on the zero-set of a holomorphic function. Electron. Commun. Probab. 13 (2008), paper no. 55, 606--613. doi:10.1214/ECP.v13-1425. https://projecteuclid.org/euclid.ecp/1465233482


Export citation

References

  • Bass, Richard F. Probabilistic techniques in analysis. Probability and its Applications (New York). Springer-Verlag, New York, 1995. xii+392 pp. ISBN: 0-387-94387-0
  • Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. xiv+464 pp. ISBN: 0-444-86172-6
  • Krantz, Steven G. Function theory of several complex variables. Second edition. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. xvi+557 pp. ISBN: 0-534-17088-9
  • Range, R. Michael. Holomorphic functions and integral representations in several complex variables. Graduate Texts in Mathematics, 108. Springer-Verlag, New York, 1986. xx+386 pp. ISBN: 0-387-96259-X
  • Daniel Revuz and Marc Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1991.