ANALYSIS ON GAUSSIAN SPACES

by Yaozhong Hu

Overview

'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.

Product Details

ISBN-13:	9789813142190
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	08/30/2016
Sold by:	Barnes & Noble
Format:	eBook
Pages:	484
File size:	78 MB
Note:	This product may take a few minutes to download.

Table of Contents

Preface vii

1 Introduction 1

2 Garsia-Rodemich-Rumsey Inequality 7

2.1 Metric Garsia-Rodemich-Rumsey Inequality 7

2.2 Volume Metric Garsia-Rodemich-Rumsey Inequality 11

2.3 Sample Path Holder Continuity of Random Fields 16

3 Analysis With Respect to Gaussian Measure in R^d 19

3.1 Gaussian Measure in R^d 19

3.2 Some Inequalities Related to Gaussian Measure 21

3.3 Brunn-Minkowski Inequality 27

3.4 Hermite Polynomials 35

3.5 Spectral Gap and Logarithmic Sobolev Inequalities 43

3.6 Variance Identity and Inequality 47

3.7 Correlation Inequality 48

3.8 Hypercontractivity 51

3.9 Hermite Polynomials in Physics and Hermite Functions 54

3.10 Segal-Bargmanu Space and Complex Hermite Polynomials 58

3.11 Segal-Bargmann Transform 63

4 Gaussian Measures on Banach Space 67

4.1 Random Variables in Banach Space 67

1.1 Abstract Wiener Space 80

4.1 Canonical Wiener Space 88

4.4 Right Tail Estimate 90

4.5 Small Ball (Left Tail) Estimate 96

5 Nonlinear Functional on Abstract Wiener Space 103

5.1 Fock Space and Chaos Expansion 103

5.2 Polarization 112

5.3 Multiple Wiener-Ito Integrals 117

5.4 Multiple Stratonovich Integrals 126

5.5 Right Tail Estimate for Homogeneous Chaos 133

5.6 Chaos Expansion of Exit Time and Skeleton of Wiener Functional 139

6 Analysis of Nonlinear Wiener Functionals 153

6.1 Gross-Sobolev Derivatives 153

6.2 Divergence Operator 163

6.3 Regularity of Density of Wiener Functional 169

6.4 Girsanov Transformation: Finite Dimension 177

6.5 Girsanov-Ramer Theorem in Abstract Wiener Space 182

6.6 Wick Product 187

6.7 Wick Renormalizatiou 194

6.8 (Noncommutative) Composition of Wiener Functional 201

6.9 Stop Brownian Motion at Anticipative Exit Time 211

7 Some Inequalities 219

7.1 Complex Hypercontractivity 219

7.2 Meyer's inequality 231

7.3 Multiplier Theorem 236

7.4 Littlewood-Paley-Stein-Meyer Theory 241

7.5 Meyer's Inequalities Revisited 257

7.6 Interpolation Inequality 265

7.7 Grothendieck Inequality 268

8 Convergence in Density 273

8.1 General Nonlinear Wiener Functional 273

8.2 Weak Convergence 276

8.3 Representation of the Derivatives of the Density 280

8.4 Random Variables in the q-th Wiener Chaos 290

8.5 Uniform Estimation of Difference of Derivatives of Densities 292

8.6 Density Convergence of Higher Bank Hermite Polynomials 297

9 Local Time and (Self-) Intersection Local Time 311

9.1 Local Time of Brownian Motion 311

9.2 Chaos Expansion of Self-inter sec lion Local Time 314

9.3 Exponential Integrability 320

9.4 Renormalizatiou When d ≠ 3 324

9.5 L2-Modulus of Continuity of Local Time of Brownian Motion 329

10 Stochastic Differential Equation 341

10.1 Existence, Uniqueness and Non-explosion 341

10.2 Hörmander Theorem 347

10.3 Exponential Integrability 364

10.4 Itô Wiener Chaos Expansion 371

10.5 FKG Inequality and Variance Inequality 373

10.6 Hypercontractivity, Spectral and Logarithmic Sobolev Inequality 376

10.7 Convergence to Density for Eigodie Diffusion 384

11 Numerical Approximation of Stochastic Differential Eequation 395

11.1 L^P Convcrgence Rate 395

11.2 Convergence in D_β,p and Convergence in Density 403

11.3 Weak Convergence Rate 417

11.4 Wong-Zakai Approximation 421

Appendix A Appendix 427

A.1 Some Elementary Results from Analysis 427

A.2 Martingales 439

Bibliography 453

Index 467