Analysis On Gaussian Spaces

Author : Yaozhong Hu
Publisher : World Scientific
Page : 484 pages
File Size : 40,5 Mb
Release : 2016-08-30
Category : Mathematics
ISBN : 9789813142190

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Analysis on Gaussian Spaces by Yaozhong Hu Pdf

Analysis 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 : Liguang Liu,Jie Xiao,Dachun Yang,Wen Yuan
Publisher : Springer
Page : 108 pages
File Size : 46,5 Mb
Release : 2018-09-20
Category : Mathematics
ISBN : 9783319950402

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Gaussian Capacity Analysis by Liguang Liu,Jie Xiao,Dachun Yang,Wen Yuan Pdf

This monograph develops the Gaussian functional capacity theory with applications to restricting the Gaussian Campanato/Sobolev/BV space. Included in the text is a new geometric characterization of the Gaussian 1-capacity and the Gaussian Poincaré 1-inequality. Applications to function spaces and geometric measures are also presented. This book will be of use to researchers who specialize in potential theory, elliptic differential equations, functional analysis, probability, and geometric measure theory.

Author : H.-H. Kuo
Publisher : Springer
Page : 230 pages
File Size : 42,7 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540375081

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Gaussian Measures in Banach Spaces by H.-H. Kuo Pdf

Author : Alexander Kukush
Publisher : John Wiley & Sons
Page : 272 pages
File Size : 53,9 Mb
Release : 2020-02-26
Category : Mathematics
ISBN : 9781786302670

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Gaussian Measures in Hilbert Space by Alexander Kukush Pdf

At 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 : Vidyadhar S. Mandrekar,Leszek Gawarecki
Publisher : CRC Press
Page : 201 pages
File Size : 44,6 Mb
Release : 2015-06-23
Category : Mathematics
ISBN : 9781498707824

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Stochastic Analysis for Gaussian Random Processes and Fields by Vidyadhar S. Mandrekar,Leszek Gawarecki Pdf

Stochastic Analysis for Gaussian Random Processes and Fields: With Applications presents Hilbert space methods to study deep analytic properties connecting probabilistic notions. In particular, it studies Gaussian random fields using reproducing kernel Hilbert spaces (RKHSs).The book begins with preliminary results on covariance and associated RKHS

Author : Wilfredo Urbina-Romero
Publisher : Springer
Page : 477 pages
File Size : 52,9 Mb
Release : 2019-06-21
Category : Mathematics
ISBN : 9783030055974

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Gaussian Harmonic Analysis by Wilfredo Urbina-Romero Pdf

Authored by a ranking authority in Gaussian harmonic analysis, this book embodies a state-of-the-art entrée at the intersection of two important fields of research: harmonic analysis and probability. The book is intended for a very diverse audience, from graduate students all the way to researchers working in a broad spectrum of areas in analysis. Written with the graduate student in mind, it is assumed that the reader has familiarity with the basics of real analysis as well as with classical harmonic analysis, including Calderón-Zygmund theory; also some knowledge of basic orthogonal polynomials theory would be convenient. The monograph develops the main topics of classical harmonic analysis (semigroups, covering lemmas, maximal functions, Littlewood-Paley functions, spectral multipliers, fractional integrals and fractional derivatives, singular integrals) with respect to the Gaussian measure. The text provide an updated exposition, as self-contained as possible, of all the topics in Gaussian harmonic analysis that up to now are mostly scattered in research papers and sections of books; also an exhaustive bibliography for further reading. Each chapter ends with a section of notes and further results where connections between Gaussian harmonic analysis and other connected fields, points of view and alternative techniques are given. Mathematicians and researchers in several areas will find the breadth and depth of the treatment of the subject highly useful.

Author : Svante Janson
Publisher : Cambridge University Press
Page : 358 pages
File Size : 43,9 Mb
Release : 1997-06-12
Category : Mathematics
ISBN : 9780521561280

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Gaussian Hilbert Spaces by Svante Janson Pdf

This book treats the very special and fundamental mathematical properties that hold for a family of Gaussian (or normal) random variables. Such random variables have many applications in probability theory, other parts of mathematics, statistics and theoretical physics. The emphasis throughout this book is on the mathematical structures common to all these applications. This will be an excellent resource for all researchers whose work involves random variables.

Author : Giuseppe Da Prato
Publisher : Springer Science & Business Media
Page : 217 pages
File Size : 47,5 Mb
Release : 2006-08-25
Category : Mathematics
ISBN : 9783540290216

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An Introduction to Infinite-Dimensional Analysis by Giuseppe Da Prato Pdf

Based on well-known lectures given at Scuola Normale Superiore in Pisa, this book introduces analysis in a separable Hilbert space of infinite dimension. It starts from the definition of Gaussian measures in Hilbert spaces, concepts such as the Cameron-Martin formula, Brownian motion and Wiener integral are introduced in a simple way. These concepts are then used to illustrate basic stochastic dynamical systems and Markov semi-groups, paying attention to their long-time behavior.

Author : A. A. Dorogovtsev
Publisher : Walter de Gruyter GmbH & Co KG
Page : 116 pages
File Size : 46,9 Mb
Release : 2019-01-14
Category : Mathematics
ISBN : 9783110618143

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Stochastic Analysis and Random Maps in Hilbert Space by A. A. Dorogovtsev Pdf

This book is devoted to stochastic operators in Hilbert space. A number of models in modern probability theory apply the notion of a stochastic operator in explicit or latent form. In this book, objects from the Gaussian case are considered. Therefore, it is useful to consider all random variables and elements as functionals from the Wiener process or its formal derivative, i.e. white noise. The book consists of five chapters. The first chapter is devoted to stochastic calculus and its main goal is to prepare the tools for solving stochastic equations. In the second chapter the structure of stochastic equations, mainly the structure of Gaussian strong linear operators, is studied. In chapter 3 the definition of the action of the stochastic operator on random elements in considered. Chapter 4 deals with the mathematical models in which the notions of stochastic calculus arise and in the final chapter the equation with random operators is considered.

Author : Daniel W. Stroock
Publisher : Unknown
Page : 0 pages
File Size : 50,5 Mb
Release : 2023
Category : Electronic
ISBN : 3031231236

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Gaussian Measures in Finite and Infinite Dimensions by Daniel W. Stroock Pdf

This 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 : Anonim
Publisher : Academic Press
Page : 424 pages
File Size : 47,7 Mb
Release : 1972-10-16
Category : Mathematics
ISBN : 0080873634

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Measure and Integration Theory on Infinite-Dimensional Spaces by Anonim Pdf

Measure and Integration Theory on Infinite-Dimensional Spaces

Author : Nobuaki Obata
Publisher : Springer
Page : 195 pages
File Size : 52,9 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540484110

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White Noise Calculus and Fock Space by Nobuaki Obata Pdf

White Noise Calculus is a distribution theory on Gaussian space, proposed by T. Hida in 1975. This approach enables us to use pointwise defined creation and annihilation operators as well as the well-established theory of nuclear space.This self-contained monograph presents, for the first time, a systematic introduction to operator theory on fock space by means of white noise calculus. The goal is a comprehensive account of general expansion theory of Fock space operators and its applications. In particular,first order differential operators, Laplacians, rotation group, Fourier transform and their interrelations are discussed in detail w.r.t. harmonic analysis on Gaussian space. The mathematical formalism used here is based on distribution theory and functional analysis , prior knowledge of white noise calculus is not required.

Author : Paul Malliavin
Publisher : Springer
Page : 346 pages
File Size : 55,7 Mb
Release : 2015-06-12
Category : Mathematics
ISBN : 9783642150746

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Stochastic Analysis by Paul Malliavin Pdf

In 5 independent sections, this book accounts recent main developments of stochastic analysis: Gross-Stroock Sobolev space over a Gaussian probability space; quasi-sure analysis; anticipate stochastic integrals as divergence operators; principle of transfer from ordinary differential equations to stochastic differential equations; Malliavin calculus and elliptic estimates; stochastic Analysis in infinite dimension.

Author : Palle Jorgensen,James Tian
Publisher : World Scientific
Page : 253 pages
File Size : 41,9 Mb
Release : 2021-01-15
Category : Mathematics
ISBN : 9789811225796

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Infinite-dimensional Analysis: Operators In Hilbert Space; Stochastic Calculus Via Representations, And Duality Theory by Palle Jorgensen,James Tian Pdf

The purpose of this book is to make available to beginning graduate students, and to others, some core areas of analysis which serve as prerequisites for new developments in pure and applied areas. We begin with a presentation (Chapters 1 and 2) of a selection of topics from the theory of operators in Hilbert space, algebras of operators, and their corresponding spectral theory. This is a systematic presentation of interrelated topics from infinite-dimensional and non-commutative analysis; again, with view to applications. Chapter 3 covers a study of representations of the canonical commutation relations (CCRs); with emphasis on the requirements of infinite-dimensional calculus of variations, often referred to as Ito and Malliavin calculus, Chapters 4-6. This further connects to key areas in quantum physics.

Author : Kazufumi Ito,Takeyuki Hida
Publisher : World Scientific
Page : 450 pages
File Size : 50,7 Mb
Release : 1991-11-29
Category : Electronic
ISBN : 9789814569378

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Gaussian Random Fields - Proceedings Of The Third Nagayo Levy Seminar by Kazufumi Ito,Takeyuki Hida Pdf

These proceedings emphasize new mathematical problems discussed in line with white noise analysis. Many papers deal with mathematical questions arising from actual phenomena. Various applications to stochastic differential equations, quantum field theory, functional integration such as Feynman integrals, limit theorems in probability are also discussed.