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A guided tour through excursions. (English) Zbl 0689.60075

The title of this survey paper is an apt description of its contents. It concentrates primarily on the theory of excursions of Brownian motions, guiding the reader without too many technicalities through the main ideas, with a few of the simpler proofs and outlines of a few of the more technical ones. The Poisson point process of excursions parametrized by local time is introduced informally and the Markovian structure of the excursion measure is analyzed in terms of its entrance law and its resolvent.
Excursions are shown to provide a powerful approach to results such as the Ray-Knight theorem, D. Williams’s path decomposition and the arc-sine law. The resolvent method for recovering a process from an excursion entrance law is illustrated by means of several examples and there is a brief section on excursions from a set. The reader is also offered firm guidance through the literature.
Reviewer: F.Papangelou

MSC:

60J65 Brownian motion
60J25 Continuous-time Markov processes on general state spaces
60G17 Sample path properties
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