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December, 1979 Brownian Motion and Analytic Functions
Burgess Davis
Ann. Probab. 7(6): 913-932 (December, 1979). DOI: 10.1214/aop/1176994888


This paper is mostly expository and is concerned with the connection between two dimensional Brownian motion and analytic functions provided by Levy's result that, if $Z_t, 0 \leqslant t < \infty$, is two dimensional Brownian motion, and if $f$ is analytic and not constant, then $f(Z_t), 0 \leqslant t < \infty$, is also two dimensional Brownian motion, perhaps moving at a variable speed. This can be used to study Brownian motion via analytic functions and, conversely, to treat analytic functions probabilistically. Recently several open problems in analytic function theory have been solved in this manner. We will present some of Doob's earlier work on the range and boundary values of analytic functions, the probabilistic theory of $H^p$ spaces due to Burkholder, Gundy and Silverstein, the author's results on conjugate function inequalities. and sketch probabilistic proofs of Picard's big and little theorems, and other theorems. There are some new results related to Hayman's generalization of Koebe's theorem.


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Burgess Davis. "Brownian Motion and Analytic Functions." Ann. Probab. 7 (6) 913 - 932, December, 1979.


Published: December, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0421.60072
MathSciNet: MR548889
Digital Object Identifier: 10.1214/aop/1176994888

Primary: 60J65
Secondary: 30A70 , 30A78 , 42A36 , 42A40

Keywords: 30-02 , analytic functions , Brownian motion , conjugate function , Hardy spaces , nontangental maximal function , Picard's theorem

Rights: Copyright © 1979 Institute of Mathematical Statistics


Vol.7 • No. 6 • December, 1979
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