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Book review

Author(s) Prato, G. Da
Zabczyk, J.
Title Stochastic equations in infinite dimensions
Publisher Cambridge Univ. Press
Year of publication 1992
Reviewed by Jordan Stoyanov

The present book is a new and strong evidence of the importance of the topic in the contemporary mathematics and its numerous applications. We find here a systematic and self contained presentation of the theory of stochastic evolution equations in Hilbert or Banach spaces. The authors demonstrate the natural connection with the deterministic evolution equations and in the "random case" with the necessity to involve stochastic integration and then naturally go to the stochastic evolution equations.

The material is well chosen and organized. In Part I the reader will find all the notions and facts which are used further on but also are of an independent interest (probability measures, stochastic processes and stochastic integrals in Hilbert spaces). The class of linear as well nonlinear stochastic equations is treated in details in Part II. Then in Part III the authors describe several interesting and important properties of the solutions (week and/or strong) of stochastic evolution equations. The Markov property, the Kolmogorov equation and the Girsanov theorem are among the most useful properties and statements concerning the solutions of stochastic evolution equations. Several related questions, e.g. the so-called small noise asymptotics, are considered and answered.

Finally, for readers convenience three appendices are included (linear deterministic equations, results on control theory and nuclear and Hilbert-Schmidt operators). The list of references is quite extensive and covers even more topics including such not discussed in the book. A subject index is also given.

There is no doubt that the book under review will be met with a strong interest by a wide category of readers not only such working in the theory but also many others looking for adequate stochastic models of quite complicated random phenomena in science and engineering. Yes, any good library will add this book to its collection, but a large number of scientists will make a real use of buying their individual copies of the present book.