7R3. Random Perturbation Methods with Applications in Science and Engineering. Applied Mathematical Sciences, Vol 150. - AV Skorokhod (Inst of Math, Ukrainian Acad of Sci, 3 Tereshchenkivska St, Kiev, 01601, Ukraine), FC Hoppensteadt (Syst Sci and Eng Res Center, Arizona State Univ, Tempe AZ 85287-7606), H Salehi (Dept of Stat and Probab, Michigan State Univ, E Lansing MI 48824). Springer-Verlag, New York. 2002. 488 pp. ISBN 0-387-95427-9. $79.95.
Reviewed by VD Radulescu (Dept of Math, Univ of Craiova, 13, St AI Cuza, Craiova, 1100, Romania).
The authors attempt to describe the basic methods and general principles of the theory of random perturbation, as well as some of the main applications of this theory in Mechanics, Engineering, Genetics, and Population Biology. This monograph contains a well-written rigorous mathematical treatment of this subject and deals with modern concepts such as dynamical systems, stochastic processes, stability, Markov chains, population biology, etc.
The volume consists of 12 chapters, but the structure of the book is roughly in two parts. The first seven chapters develop several mathematical methods that are useful in the study of random perturbations of dynamical systems. The second part (Chs 8–12) presents nonrandom problems in a variety of important applications.
The purpose of Chapter 1 is to describe some basic ergodic theorems (including Birkhoff’s Classical Ergodic Theorem) from a mathematical point of view and to recall some results that will be used in the book. There are also described discrete-time and continuous-time Markov processes, and continuous-time stationary processes.
Chapter 2 is devoted to some convergence properties of stochastic processes. Technical conditions that are sufficient for various kinds of convergence are described. The results contained in this chapter are developed mostly in the framework of continuous-time processes.
In Chapter 3 the authors develop averaging methods for random perturbations of Volterra integral equations, of differential equations and of difference equations. In each of these cases, the remainder term is considered in greater detail in Chapter 4. The main results in this part of the book establish that the deviation tends to zero in a prescribed sense if an averaging theorem is true.
Chapter 5 deals with randomly perturbed systems of differential and difference equations whose averaged systems are static. The main result contained in this chapter establishes that, under some reasonable assumptions, the stochastic process becomes, asymptotically, a diffusion process.
Chapter 6 gives an outlook to a variety of stability problems for differential and difference equations when they are perturbed by random noise. The authors also describe the growth of solutions to certain randomly perturbed convolution equations. The abstract results are illustrated by several examples of stability phenomena.
Chapter 7 deals with Markov chains having a finite state space, but random transition probabilities. The basic hypothesis is that the transition probabilities of the Markov chain are close to those of a homogeneous Markov chain having nonrandom transition probabilities. Under this assumption, the authors establish the asymptotic behavior of the chain and its transition probabilities.
In Chapter 8 it is described a method for studying various kinds of bifurcations for two-dimensional mechanical systems. There are also discussed random perturbations of oscillatory linear systems and of rigid-body motions.
In Chapter 9 the authors consider a dynamical system on a two-dimensional torus. The main results involve either flows that have non random elements or random perturbations of problems of this type.
In Chapter 10 an important electronic circuit, namely the phase-locked loop is described. It first investigates the circuit’s dynamics without and with random perturbations, but in the absence of external forcing. Next, the response of the circuit to noisy external signals is analyzed.
Interesting applications are also given in Chapter 11, in which the authors consider random perturbation of ecological systems, of epidemic disease processes, and of demographic models. Chapter 12 provides several concrete problems from genetics. In both chapters, the authors describe classical models for these phenomena and related results concerning their asymptotic behavior for large time.
Appendix A recalls some basic notions of probability theory, and Appendix B contains historical comments and further remarks.
In summary, this reviewer would definitely recommend Random Perturbation Methods with Applications in Science and Engineering to those researchers and graduate students in Science and Engineering who require tools to investigate stochastic systems. The book qualifies to be a reference work that certainly would be a valuable addition in libraries of universities and laboratories pursuing research in Applied Mathematics.