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Diffusions and elliptic operators. (English) Zbl 0914.60009

Probability and Its Applications. New York, NY: Springer. xiii, 232 p. (1998).
The main subject of the monograph is the connection between PDEs and diffusion processes. After an introductory chapter on stochastic differential equations (SDEs), probabilistic representations of solutions to the following PDEs as expected values of functionals of diffusions are presented: Poisson’s equation, Dirichlet problem, Cauchy problem, Schrödinger equation and Neumann and oblique derivative problems. Existence, uniqueness and smoothness of solutions to these equations are deferred to the third chapter which is of a more analytic nature. The special case of one-dimensional diffusions is treated in a separate chapter. The following three chapters contain a number of estimates related to diffusion operators with bounded coefficients in nondivergence and divergence form, e.g. Harnack’s inequality and heat kernel estimates as well as a detailed treatment of the martingale problem associated to a diffusion operator. In the final chapter the author presents two approaches to the Malliavin calculus (Girsanov transformation and Ornstein-Uhlenbeck operator) and provides a proof of Hörmander’s theorem on the existence of a smooth transition density for a (nice) diffusion.
In spite of a few inaccuracies (e.g. Theorem 7.1 in Chapter VII which does not even hold for Brownian motion) the book is well-written, readable and provides a comprehensive treatment of the subject. Even though the author primarily addresses probabilists with an interest in PDEs, analysts may like the book too. The reader should have some prior knowledge in probability theory. Basic concepts and results like martingale, Doob’s inequality and Lévy’s theorem are stated without much explanation. The chapter on stochastic differential equations however is essentially self-contained.

MSC:

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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