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Lévy processes in Lie groups. (English) Zbl 1076.60004

Cambridge Tracts in Mathematics 162. Cambridge: Cambridge University Press (ISBN 0-521-83653-0/hbk; 0-511-19262-2/e-book). x, 266 p. (2004).
Lévy processes with values in Lie groups have now been studied for a long time. Lie groups are indeed the natural structure where one can define Lévy processes. The purpose of this book is to explain recent progress which was made (mainly by the author) about the limiting properties in large time of these processes.
In the first part, the reader is introduced to these processes, in particular to their generator and to their representation as solutions of stochastic integral equations. Their behaviour in the case of compact Lie groups is also studied by means of Fourier analysis. In the second (and main) part, the reader is invited to concentrate on the large time behaviour of Lévy processes in semi-simple Lie groups of noncompact type; the study is based on decompositions of these groups which lead to decompositions of the process. A chapter gives the basic limiting properties; the rate of convergence is worked out in another chapter, and the last chapter gives applications to the study of stochastic flows (which can be viewed as Lévy processes in the diffeomorphism group). Two appendices give basic definitions and results about Lie groups and stochastic calculus. A chapter of the book also contains an introduction to semi-simple Lie groups of noncompact type.
A part of the book is rather technical, and probably not very easy to read for someone who is not familiar with Lie groups and stochastic calculus, but this cannot be avoided. The author has made everything necessary so that the book be self contained. Moreover it is well written and I found the reading of several parts of this book very pleasant. The topic is modern and interesting. Most of the results were proved recently by the author and cannot be found in other books. I think that this book should be present in the library of any researcher interested in Lévy processes and Lie groups.

MSC:

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60F15 Strong limit theorems
60G51 Processes with independent increments; Lévy processes
60J75 Jump processes (MSC2010)
22E30 Analysis on real and complex Lie groups
22E46 Semisimple Lie groups and their representations
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