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Book review

Author(s) Belopolskaya, Ya.I.
Dalecky, Yu.L.
Title Stochastic Equations and Differential Geometry
Publisher Kluwer Academic Publishers
Year of publication 1990
Reviewed by Viorica-Cerasela Postolache

In this book are set forth the fundamental notions and results concerning the theory of stochastic differential geometry and also some applications of this theory in solving Cauchy problems for partial differential equations. This work gives a sufficient number of results, so as to permit the reader to give an orientation in some new and important fields of Mathematics, Physics and Biology which are strongly related with one another.

The book includes six chapters, interesting historical comments, a very good and comprehensive bibliography, a table of subjects index as well as a table of contents sufficiently detailed. The first of these chapters deals with functions and measures in Banach spaces. The second, gives the basic elements of the theory of connections for vector bundles. In the third chapter, the theory of stochastic equations in Banach spaces is presented.

In Chapter 4, a stochastic differential equation on a smooth Banach manifold is constructed and a Cauchy problem for it is solved. In Chapter 5 is investigated the Cauchy problem for some parabolic equations (Kolmogorov type equations). In the last chapter diffusion processes on Lie Groups are treated.

This very well produced book is suitable for more advanced students in mathematics and physics, applied mathematicians and physicists as well as for scientists seriously interested in the topic of differential stochastic equations. An introductory course to differential geometry and to probability theory is assumed.

The presentation of the book is rigorous, strict proofs of the stated results are given. Many results reflects the experience of the authors in the difficult field of the stochastic differential geometry research work (see the diffusion processes in smooth Banach spaces, diffusion and quasi-invariant measures on infinite dimensional Lie groups and so on). The book is sufficiently self contained to enable independent research. Some chapters allow an independent study. Consequently a large area of readers can use it. Purchasing this book is a good investment for individuals and libraries alike.