Analysis on Gaussian Spaces

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.

Contents:

• Introduction
• Garsia–Rodemich–Rumsey Inequality
• Analysis with Respect to Gaussian Measure in ℝd
• Gaussian Measures on Banach Space
• Nonlinear Functionals on Abstract Wiener Space
• Analysis of Nonlinear Wiener Functionals
• Some Inequalities
• Convergence in Density
• Local Time and (Self-) Intersection Local Time
• Stochastic Differential Equation
• Numerical Approximation of Stochastic Differential Equation

Readership: Graduate students and researchers in probability and stochastic processes and functional analysis.
Key Features:

• The book provides a self-contained presentation of some fundamental results on analysis of Gaussian functionals. These results cannot be found in any single existing source before this book
• Author provides new and different proofs for a number of theorems (such as the proofs of logarithmic inequality, correlation inequality, hypercontractivity and so on)
• The existing proofs of some theorems are probabilistic and use Brownian motions and so on. The book has made effort to provide analytic proofs
• This book contains some recent progress in this field