Characterization of Gaussian Measure on Banach Space
Author: 范君豪
Publisher:
Published: 2008
Total Pages: 12
ISBN-13:
DOWNLOAD EBOOKRead and Download eBook Full
Author: 范君豪
Publisher:
Published: 2008
Total Pages: 12
ISBN-13:
DOWNLOAD EBOOKAuthor: Yaozhong Hu
Publisher: World Scientific
Published: 2016-08-30
Total Pages: 483
ISBN-13: 9813142197
DOWNLOAD EBOOK'Written by a well-known expert in fractional stochastic calculus, this book offers a comprehensive overview of Gaussian analysis, with particular emphasis on nonlinear Gaussian functionals. In addition, it covers some topics that are not frequently encountered in other treatments, such as Littlewood-Paley-Stein, etc. This coverage makes the book a valuable addition to the literature. Many results presented in this book were hitherto available only in the research literature in the form of research papers by the author and his co-authors.'Mathematical Reviews ClippingsAnalysis of functions on the finite dimensional Euclidean space with respect to the Lebesgue measure is fundamental in mathematics. The extension to infinite dimension is a great challenge due to the lack of Lebesgue measure on infinite dimensional space. Instead the most popular measure used in infinite dimensional space is the Gaussian measure, which has been unified under the terminology of 'abstract Wiener space'.Out of the large amount of work on this topic, this book presents some fundamental results plus recent progress. We shall present some results on the Gaussian space itself such as the Brunn-Minkowski inequality, Small ball estimates, large tail estimates. The majority part of this book is devoted to the analysis of nonlinear functions on the Gaussian space. Derivative, Sobolev spaces are introduced, while the famous Poincaré inequality, logarithmic inequality, hypercontractive inequality, Meyer's inequality, Littlewood-Paley-Stein-Meyer theory are given in details.This book includes some basic material that cannot be found elsewhere that the author believes should be an integral part of the subject. For example, the book includes some interesting and important inequalities, the Littlewood-Paley-Stein-Meyer theory, and the Hörmander theorem. The book also includes some recent progress achieved by the author and collaborators on density convergence, numerical solutions, local times.
Author: Alexander Kukush
Publisher: John Wiley & Sons
Published: 2020-02-26
Total Pages: 272
ISBN-13: 1786302675
DOWNLOAD EBOOKAt the nexus of probability theory, geometry and statistics, a Gaussian measure is constructed on a Hilbert space in two ways: as a product measure and via a characteristic functional based on Minlos-Sazonov theorem. As such, it can be utilized for obtaining results for topological vector spaces. Gaussian Measures contains the proof for Ferniques theorem and its relation to exponential moments in Banach space. Furthermore, the fundamental Feldman-Hájek dichotomy for Gaussian measures in Hilbert space is investigated. Applications in statistics are also outlined. In addition to chapters devoted to measure theory, this book highlights problems related to Gaussian measures in Hilbert and Banach spaces. Borel probability measures are also addressed, with properties of characteristic functionals examined and a proof given based on the classical Banach–Steinhaus theorem. Gaussian Measures is suitable for graduate students, plus advanced undergraduate students in mathematics and statistics. It is also of interest to students in related fields from other disciplines. Results are presented as lemmas, theorems and corollaries, while all statements are proven. Each subsection ends with teaching problems, and a separate chapter contains detailed solutions to all the problems. With its student-tested approach, this book is a superb introduction to the theory of Gaussian measures on infinite-dimensional spaces.
Author: Daniel Li
Publisher: Cambridge University Press
Published: 2018
Total Pages: 405
ISBN-13: 1107162629
DOWNLOAD EBOOKThis second volume of a two-volume overview focuses on the applications of Banach spaces and recent developments in the field.
Author: H.-H. Kuo
Publisher: Springer
Published: 2006-11-14
Total Pages: 230
ISBN-13: 3540375082
DOWNLOAD EBOOKAuthor: James Kuelbs
Publisher:
Published: 1972
Total Pages: 29
ISBN-13:
DOWNLOAD EBOOKUsing the theory of operator-valued reproducing kernels a necessary and sufficient condition for equivalence or singularity of two Gaussian measures corresponding to a Banach space-valued stochastic processes is given. The characterization is in terms of operator-valued covariance kernels associated with these measures. The result is applied to the Wiener process with a Banach state space and an infinite dimensional extension of a result of Shepp is obtained. (Author).
Author: Vladimir Igorevich Bogachev
Publisher: American Mathematical Soc.
Published: 1998
Total Pages: 449
ISBN-13: 0821810545
DOWNLOAD EBOOKThis 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.
Author: Daniel W. Stroock
Publisher:
Published: 2023
Total Pages: 0
ISBN-13: 9783031231230
DOWNLOAD EBOOKThis text provides a concise introduction, suitable for a one-semester special topics course, to the remarkable properties of Gaussian measures on both finite and infinite dimensional spaces. It begins with a brief resumé of probabilistic results in which Fourier analysis plays an essential role, and those results are then applied to derive a few basic facts about Gaussian measures on finite dimensional spaces. In anticipation of the analysis of Gaussian measures on infinite dimensional spaces, particular attention is given to those properties of Gaussian measures that are dimension independent, and Gaussian processes are constructed. The rest of the book is devoted to the study of Gaussian measures on Banach spaces. The perspective adopted is the one introduced by I. Segal and developed by L. Gross in which the Hilbert structure underlying the measure is emphasized. The contents of this book should be accessible to either undergraduate or graduate students who are interested in probability theory and have a solid background in Lebesgue integration theory and a familiarity with basic functional analysis. Although the focus is on Gaussian measures, the book introduces its readers to techniques and ideas that have applications in other contexts.
Author: J.-A. Chao
Publisher: Springer
Published: 2006-11-17
Total Pages: 238
ISBN-13: 354039284X
DOWNLOAD EBOOKAuthor: Daniel Li
Publisher: Cambridge University Press
Published: 2018
Total Pages: 463
ISBN-13: 1107160510
DOWNLOAD EBOOKThis first volume of a two-volume overview covers the basic theory of Banach spaces, harmonic analysis and probability.