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Book review

Author(s) Skorohod, A.V.
Title Stochastic equations for complex systems
Publisher Reidel/Kluwer
Year of publication 1988
Reviewed by Günter Vojta

The theories of Brownian motion and of Markov processes have been in the focus of scientific interest for many decades. In this volume the well-known author presents new studies on this topic employing a new method, viz. a theory of infinite systems of linear stochastic differential equations. The result is a research monograph on a highly specialized level, full of mathematical rigour, but well readable due to a lot of motivations, explanations, historical remarks, and other background information. It is a good translation of the Russian original of 1983.

The notion of complex systems means systems to be described in phase spaces of varying or infinite dimensions or of locally non-Euclidean character. The book is divided into two large chapters. In the first one continuous Markov processes in a locally compact space are considered using modern tools like the martingale concept and semigroup theory. Here questions of existence and uniqueness of solutions of stochastic equations are in the foreground as well as the corresponding limit theorems. The second chapter contains a stochastic theory for systems of randomly interacting essentially classical particles where the complexity is caused by the geometric shape of the particles and the presence of internal degrees of freedom, by different external and interparticle forces, and eventually by quantum mechanical effects. Again problems of the asymptotic behaviour and limit theorems for statistical distribution functions are treated. In particular, fluctuations and the diffusion approximation for the particle transport are thoroughly discussed. There are a short bibliography and a subject index.

In his preface the author emphasizes that the book is of a purely mathematical nature. Nevertheless, the work will be of great interest not only to mathematicians but also to physicists working in the fields of statistical thermodynamics and stochastic processes. Unfortunately, no reference is made to the theory of I. Prigogine (Brussels) on the general solution of the N-particle Liouville equation nor to the general (non-Markovian) theory of Brownian motion of H.Mori (Kyoto, now Fukuoka), both physical theories being burdened by mathematical difficulties which seem to be overcome by the present author - at least in a certain sense and at least partially.

The book is well produced and can be recommended to all scientists and graduate students interested in the fields of stochastics and mathematical or theoretical physics. Libraries should purchase it.