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 Electronic Communications in Probability > Vol. 13 (2008) > Paper 55 open journal systems 


A martingale on the zero-set of a holomorphic function

Peter Kink, Faculty for Computer and Information Sciences, University of Ljubljana


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.


Full text: PDF

Pages: 606-613

Published on: November 24, 2008
Bibliography
  1. Daniel Revuz and Marc Yor. Continuous martingales and Brownian motion. Springer-Verlag 1991
  2. Nobuyuki Ikeda and Shinzo Watanabe. Stochastic Differential Equations and Diffusion Processes. North Holland Publishing Company 1991.
  3. Richard F. Bass. Probabilistic Techniques in Analysis. Springer-Verlag 1995
  4. Steven George Krantz. Function theory of several complex variables. Pacific Grove : Wadworth and Brooks 1992
  5. Michael Range. Holomorphic functions and integral representations in several complex variables. Springer-Verlag 1986
















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Electronic Communications in Probability. ISSN: 1083-589X