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An introduction to random currents and their applications. (English) Zbl 1426.28001

SpringerBriefs in Mathematics. Cham: Springer (ISBN 978-3-319-94576-7/pbk; 978-3-319-94577-4/ebook). xiii, 143 p. (2018).
Typically for this series, the present book covers in a combined way topics from a few areas of mathematics such as measure theory, functional analysis, geometry and probability. One of the essential ingredients is that the main objects studied are random, thus most of the questions and their answers are given in probabilistic language.
The book starts with a special and impressive foreword written by Willi Jäger (Heidelberg). He explains the difficulties when studying real world phenomena which are both complex and random. Hence, the necessity to attract different mathematical ideas and tools for solving diverse non-trivial problems.
The goal of the author is to introduce and describe objects called random currents. Just think of random measures or random functionals. Attention is paid to both aspects, theoretical and applied.
The first about 60 pages form the main part of the book. Let us list the names of the chapters and indicate in brackets key words and phrases:
1.
Introduction and motivation.
2.
Differential forms (\(m\)-forms, operations on differential forms, line integrals, Stokes theorem).
3.
Currents: the deterministic case (distributions, \(m\)-currents, operations, Lie derivatives).
4.
Currents: the stochastic case (random Radon measures, random closed sets, continuity, random currents).
5.
Applications (tumor-driven angiogenesis, capillary networks, crystal dislocations, Gaussian currents, shape analysis, space of currents on a RKHS).
The next part of about 60 pages consists of five useful appendices: A. Elements of measure theory. B. Fundamentals of stochastic processes. C. Vector calculus. D. Regular surfaces. E. Reproducing kernel Hilbert spaces. There are also glossary, references and index.
All notions considered in the book are well defined, some of their main properties are formulated with brief comments. Usually the reader is referred to an appropriate source for details. Thus, a reader with serious background in a few branches of mathematics will be successful after reasonable additional work.
Even following a compact style of presentation, the author is successful in introducing the reader to a relatively new area of study. It is clear from the text that there are new mathematical questions coming out and new questions coming from other areas.
The book will be useful to mathematicians and applied scientists. Hopefully, in the future, there will be more books of larger size with more details of both theoretical and applied nature.

MSC:

28-02 Research exposition (monographs, survey articles) pertaining to measure and integration
28A75 Length, area, volume, other geometric measure theory
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60D05 Geometric probability and stochastic geometry
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