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Probabilistic techniques in analysis. (English) Zbl 0817.60001
New York, NY: Springer-Verlag. xii, 392 p. (1995).
This book is a wellcome modern survey on the rather nonstandard subject - - the influence of the probabilistic ideas on analysis, especially on potential theory, singular integrals and complex analysis. It will be interesting for specialists in both fields: in theory of probability and analysis. Also this book may be useful for beginning Ph. D. students because it includes not only the recent developments, but proofs of all auxiliary results from probability theory and analysis. The book consists of 5 chapters.
Chapter 1 is a minicourse in stochastic analysis including the main properties of Brownian motion, a brief account of Markov processes, semigroups and local times, the elements of the martingale theory, an introduction to stochastic differential equations. Chapter 2 treats the connection between potential theory and Brownian motion, in particular the behaviour of harmonic functions in domains with smooth boundary, the solution of the Dirichlet problem, Newton potentials and Green functions, Riesz decomposition and the construction of the Martin boundary. Chapter 3 deals with the behaviour of harmonic functions in the wider class of Lipschitz domains. A good number of these results belongs to the author (together with Burdzy). The author gives probabilistic proofs of the boundary Harnack principle, which occurs to be useful for the construction of the Martin boundary and proofs of Fatou theorems for Lipschitz domains. In Chapter 4 the properties of the martingale transform of a plane Brownian motion are applied to obtain many results about Hilbert and Riesz transforms, Littlewood-Paley functions and some other singular integrals. Chapter 5 is devoted to analytic functions and conformal mappings. The properties of 2-dimensional Brownian motion are used to prove such important theorems as Riesz-Thorin interpolation and Riemann’s mapping theorems, Picard’s little theorem and Koebe’s distortion theorem, Makarov’s LIL for Bloch functions. The boundary behaviour of analytic functions and the supports of harmonic measures are investigated. Finally, the probabilistic Varapoulos proof of the famous corona problem is proposed.
Each chapter concludes with exercises ranging from routine to difficult ones. Also at the end of the last three chapters a number of interesting open problems are discussed.

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60J65 Brownian motion
31C15 Potentials and capacities on other spaces
42A61 Probabilistic methods for one variable harmonic analysis
44A15 Special integral transforms (Legendre, Hilbert, etc.)
42B25 Maximal functions, Littlewood-Paley theory
60J45 Probabilistic potential theory