Gaussian Measures (Mathematical Surveys and Monographs) by Vladimir I. Bogachev (PDF)

Description

This book gives a systematic exposition of the modern theory of Gaussian measures. It presents with complete and detailed proofs fundamental facts about finite and infinite dimensional Gaussian distributions. Covered topics include linear properties, convexity, linear and nonlinear transformations, and applications to Gaussian and diffusion processes. Suitable for use as a graduate text and/or a reference work, this volume contains many examples, exercises, and an extensive bibliography. It brings together many results that have not appeared previously in book form.

User’s Reviews

Editorial Reviews: Review A deep and detailed treatment of an important subject. –American Mathematical MonthlyThe scope of topics related to Gaussian processes and measures is huge. Complemented by a list of problems in each chapter, a collection of over 800 references and a detailed section of bibliographical comments, Bogachev’s book covers, in varying degrees of detail, an impressive portion of the theoretical part of this scope. It does so in a clear and precise mathematical style. –Mathematical Reviews About the Author Vladimir I. Bogachev , Moscow State University, Russia.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The author is a mathematician from Moscow who specialises in measure theory. It is a book on advanced mathematics – you can read the inside of it. Anyway, the book has been well written, with lots of exercises and copious historical notes. I can also recommend his two-volume treatise “Measure Theory”. If your library has Halmos’ classic work then you might consider adding these works by Bogachev as well. They will be classics.

⭐Most of the book is on Gaussian measures generally on locally convex spaces. From the scattered reading I’ve done, that there are few expositions on this. Although Bogachev is writing about machinery with many moving parts, he writes in a nice pedestrian way, and rather than asserting things to be trivial that are messy to write out he either gets his hands dirty or cites a proof, which at least lets the reader know that the result is not meant to be obvious. In the second chapter Bogachev talks about the Cameron-Martin space. However, because locally convex spaces are not as securely in the arsenal of analysts as they ought to be, I would like the appendix to be expanded with proofs of some of the results that are stated about the Borel sigma-algebra of a locally convex space, especially because so many of the references are to Russian books.I have spent serious time with the first chapter of this book, which is about Gaussian measures on d-dimensional Euclidean space. This can be read on its own, and is closer to what is commonly talked about in probability theory than are measures on locally convex spaces. In probability theory one speaks about random variables with a multivariate normal distribution, and this means that their pushforward measure is a d-dimensional Gaussian measure, and working just with measures on d-dimensional Euclidean space is a better setting for proving some things than working through random variables. This chapter has the best presentation of Hermite polynomials, the Ornstein-Uhlenbeck semigroup (which Bogachev defines using the Mehler formula), and the logarithmic Sobolev inequality I have seen in any book in probability theory, partial differential equations, or functional analysis.

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