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Probability methods in the theory of conformal mappings. (English. Russian original) Zbl 0736.30006
Leningr. Math. J. 1, No. 1, 1-56 (1990); translation from Algebra Anal. 1, No. 1, 3-59 (1989).
This paper is a survey of recent results on Bloch functions and the boundary behaviour of conformal maps of the unit disk into the plane. Starting in 1984 the author has solved several important problems that had been open for many years. The present paper also contains a number of results not published before. The paper presents many proofs. Some are sketched while others appear here for the first time in detail. The methods come from probability (in particular martingale) theory and complex analysis. The commented section headings give some idea of the wealth of material:
1. Bloch class: Analytic functions \(f\) with \(\hbox{sup}(1-| z|)| f'(z)|<\infty\) and their connection to conformal maps and geometry.
2. Bloch functions and dyadic martingales: The martingale associated with a Bloch function and martingale inequalities with applications to conformal maps.
3. Law of the iterated logarithm: The precise a.e. rate of growth of Bloch functions and of the derivative of conformal maps.
4. The angular derivative problem and random walk on the line: Existence of the angular derivative of conformal maps and the compression of Hausdorff measures.
5. Random walk in the plane associated with an almost conformal martingale: Radial boundedness of Bloch functions.
6. First passage problems for martingales and boundary distortion: The absolute continuity and singularity of a conformal map for various Hausdorff measures. Many of these results are new.
7. Domains with self-similar boundary and Gibbs measures.

30C35 General theory of conformal mappings
30D45 Normal functions of one complex variable, normal families
60G46 Martingales and classical analysis