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Excursions of Markov processes. (English) Zbl 0983.60504

Basel: Birkhäuser. xii, 275 p. (1992).
This book assembles a number of results in excursion theory, starting from the fundamental work of Itô and Motoo, through the developments of Maisonneuve, Williams, the reviewer, Salisbury, Le Gall, et al.; the viewpoint adopted is strictly that of the abstract theory of Markov processes, so that techniques of stochastic calculus, for example, are not used.
After an introductory chapter on Markov processes, a chapter on examples of Markov processes leads the reader gently to a characterisation of Feller Brownian motions, which illustrate most of the key ideas of Itô excursion theory. Chapter III carefully develops the classical Itô theory of the Poisson point process of excursions from a single distinguished point. Chapter IV treats the paradigm example of the Brownian excursion process. (The reader should be aware that the author’s normalisation of local time differs from both of those in common use, the Itô-McKean and the occupation density! However, the choice is natural within the strict confines of Markov process theory.)
The fifth chapter describes a pathwise construction of a Markov process from the Poisson point process of excursions, under slightly restrictive conditions. The difficulty of proving the intuitively obvious Markov property is apparent. Chapter VI proves the Ray-Knight theorem, and discusses the excursion filtration and Le Gall’s lovely construction of super-Brownian motion from a Brownian excursion. The final chapter is an account of the more difficult topic of excursions away from a set; considerable technique is needed to make precise sense of the intuitively reasonable statements which appear, and in the end there are not many examples where one can say much.

MSC:

60J25 Continuous-time Markov processes on general state spaces
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60J55 Local time and additive functionals
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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