Yaozhong Hu - Analysis on Gaussian Spaces

Yaozhong Hu University of Kansas , USA Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Hu, Yaozhong, 1961 Title: Analysis on Gaussian spaces / by Yaozhong Hu (University of Kansas, USA). Description: New Jersey : World Scientific, 2016. |
Includes bibliographical references and index.

Identifiers: LCCN 2016025210 | ISBN 9789813142176 (hardcover : alk. paper) Subjects: LCSH: Spaces of measures. | Gaussian measures. | Measure theory. | Gaussian distribution. | Distribution (Probability theory) Classification: LCC QA312 .H8 2016 | DDC 515/.42--dc23 LC record available at https://lccn.loc.gov/2016025210 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

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Printed in Singapore To Jun Fu and To Ruilong Hu

To study the classical analysis, one needs to understand well the Euclidean space d and the Lebesgue measure. One expects the same when one studies the nonlinear Gaussian functionals. The study of infinite dimensional Banach space with a Gaussian measure (Gaussian space) and analysis of nonlinear functionals on Gaussian space have been studied since long and have found many applications. However, they are usually in different books aiming at different communities. One of the goals of this book is to put them together. It intends to present some basic materials on Gaussian space and on analysis of nonlinear Gaussian functionals.

It differs from some existing books on Malliavin calculus not only because it contains some material on Gaussian space but also because it contains some material on analysis of nonlinear Gaussian functionals which are absent in popular Malliavin calculus books. For example, this book includes some important topics such as Littlewood-Paley-Stein-Meyer theory, interpolation inequality, complex hypercontractive inequality, correlation inequality, polarization technique, Wick product and so on. Effort has been made to allow this book accessible to broader audience. For example, some probabilistic proofs of (complex) hypercontractivity, correlation inequality and so on are now given in purely analytic ways. To facilitate the reading of the book, an appendix is included to present some basic results and their proofs from analysis and probability. Due to page limitation the book excludes many other important applications such as applications to fractional Brownian motions, stochastic partial differential equations, and mathematical finance.

I take this opportunity to thank my advisor, P. A. Meyer, who taught me probability from the beginning. Although I had had a master degree in system science before I went to study under his guidance, I had very limited knowledge on probability theory back then. I am so grateful for his endless patience and generous encouragement. Many mathematicians have helped me through my career of study and research.

I very much appreciate their open-handed help and constant support. Here, let me mention particularly my postdoctoral advisors: Sergio Albeverio, Gopinath Kallianpur, Bernt ksendal, and Weian Zheng. I also thank my colleagues, Tyrone Duncan, David Nualart, and Bozenna Pasik-Duncan. Special thanks go to David Elworthy and Jiaan Yan. Finally, I would like express my sincere appreciation to Ms Rok Ting Tan and other staff from World Scientific for their generous support in the publication of this book. Yaozhong Hu
Lawrence, Kansas
and Oslo, Norway

Analysis of functions on finite dimensional Euclidean space d with Lebesgue measure dx is one of the most important mathematical subjects.

Researchers have been working to extend it to infinite dimension. In the absence of measure, there have been a great amount of works on nonlinear functional analysis, see for example, [Zeidler (1986, 1990a,b, 1985, 1988)]. Infinite dimensional Lebesgue measure has been (formally) used in physics such as in the famous Feynman path integral, which is one of the backbones of quantum physics. However, it is well-known (see ) that in infinite dimensional space there is no Lebesgue measure. The next natural choice for infinite dimensional measure is the Gaussian measure, which is a family of measures depending on the mean and covariance. In fact, there have been large amount of works to justify the Feynman path integral by using analytic extension of integrals with respect to Gaussian measures.

Another important application to infinite dimensional Gaussian measure is the Euclidean quantum field theory (see [Glimm and Jaffe (1987); Simon (1974, 2005)]) and stochastic quantization (see [Hu and Kallianpur (1998)] and the references therein). However, even in finite dimensional case, analysis with respect to Gaussian measure is not a trivial analogy of the classical analysis with respect to Lebesgue measure. Many most frequently used properties with respect to Lebesgue measure is no longer true for Gaussian measure. For example, Lebesgue measure is translation invariant. Namely, if B is a Borel measurable subset of d, then B and a + B = {a + b, bB} have the same Lebesgue measure. But this is no longer true for Gaussian measure, which concentrates around the mean.

However, it is quasi-invariant in the sense that Girsanov theorems hold (see details the Gaussian measure itself. There have been several books on Gaussian measure ([Bogachev (1998); Kuo (1975); Ledoux and Talagrand (2011); Lifshits (2012)]). In this book, we shall first present some properties of Gaussian measures. In particular, we shall present the Brunn-Minkowski inequality, spectral gap and logarithmic Sobolev inequality (and their unification), hypercontractive inequality, variance inequality, correlation inequality (see Section 3). Usually, if an inequality holds true in finite dimension and if the constants appeared in the inequality do not depend on dimension, then this inequality can be extended to infinite dimension (via finite dimensional approximation). Their forms are straightforward and we shall not repeat them.

In probability theory, a Gaussian measure is usually given by a Gaussian stochastic process. The Gaussian measure can also be characterized by the corresponding process. As for general stochastic process a basic property is the sample path Hlder continuity. This amounts to say that the Gaussian measure is supported by the space of Hlder continuous functions. In this direction, the Garsia-Rodemich-Rumsey inequality seems to be quite powerful. We present this inequality and its extension from metric to volumetric case in Section 2.