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.
"A martingale on the zero-set of a holomorphic function." Electron. Commun. Probab. 13 606 - 613, 2008. https://doi.org/10.1214/ECP.v13-1425