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Nonstandard methods in stochastic analysis and mathematical physics. (English) Zbl 0605.60005
Pure and Applied Mathematics, Vol. 122. Orlando etc.: Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers). xi, 514 p. Cloth: $ 59.50; Paper: $ 34.50 (1986).
In this very significant book, the authors who are well known authorities in this particular area of nonstandard analysis, present many of the most useful aspects of the nonstandard approach to stochastic analysis. The tone of this volume is towards a more intuitive view of these processes as they apply to physical models.
This book is relatively self-contained in that it first presents the customary, and to this reviewer’s mind the best, ultraproduct construction for the nonstandard universe of internal objects relative to a superstructure. Indeed, the first chapter is a minicourse in the foundations of nonstandard analysis as it has been developed by those that follow the ideas first put forth by Robinson. This fact alone is significant. Each chapter has an extensive list of references that includes many of the most recent contributions to this important area. The exposition is excellent and motivates the concepts involved with additional remarks and examples.
The volume should have a wide ranging appeal to the physics community. However, I am not sure whether or not they are willing to give up their less intuitive methods and adopt the nonstandard approach unless this is the only approach to which they have been exposed. In the reviewer’s opinion the continual re-establishing of standard results by nonstandard methods may not produce an immediate ”rush” to adopt nonstandard analysis within the applied areas. On the other hand, if there exists an objectively real nonstandard world that can only be described by the behavior of pure internal or external objects - a world that better explains the behavior of standard physical world, then nonstandard analysis would not only be intuitively useful but would be necessary.
In order to indicate the usefulness of this volume, which is not a text, to the scientific community we list its contents chapter by chapter. Chapter 1 includes the customary approach to the construction of the extended universe, limits, continuity, the derivative, ordinary integrals and differential equations. Chapter 2 discusses certain aspects of topological spaces, minor considerations of saturation, linear spaces, spectral decomposition and nonstandard methods on Banach spaces. Chapter 3 is a general account of the nonstandard approach to probability theory with a few applications to Brownian motion. Note that chapters 1 to 3 are very useful since they are quite general in character.
The second part of this volume presents selected applications of the previous methods. Chapter 4 makes an extensive study of stochastic analysis ranging from stochastic integration, stochastic differential equations through white noise and Levy Brownian motion. This chapter also includes martingale integration. Chapter 5 discusses hyperfinite Dirichlet forms and Markov processes. This includes applications to quantum mechanics. In particular, Hamiltonians, standard and nonstandard energy forms, and Markov processes are significant. Chapter 6 deals with certain topics in differential operations. These include singular perturbations of non-negative operators, point interactions, perturbations of local time functionals and applications to the Boltzmann equation. We even find in this chapter an attempt to clear up some of the difficulties with constructing an acceptable and rigorous approach to Feynman path integrals. Finally chapter 7 investigates the stochastic evolution of lattice systems. Within this chapter we find a nonstandard approach to the global Markov property, the interesting concepts of the hyperfinite models for quantum field theory and polymers.
Certainly this volume should be part of every library collection on the subject matter of stochastic processes.

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60G05 Foundations of stochastic processes
60H99 Stochastic analysis
03H10 Other applications of nonstandard models (economics, physics, etc.)
60G44 Martingales with continuous parameter