Electronic Communications in Probability

A martingale on the zero-set of a holomorphic function

Peter Kink

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

complex martingales stochastic differential equations

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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

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