Gaussian Measures in Finite and Infinite Dimensions

ISBN-13: 9783031231216 / Angielski / Miękka / 2023 / 144 str.

Daniel W. Stroock

Opis

This text provides a concise introduction, suitable for a one-semester special topicscourse, to the remarkable properties of Gaussian measures on both finite and infinitedimensional spaces. It begins with a brief resumé of probabilistic results in which Fourieranalysis plays an essential role, and those results are then applied to derive a few basicfacts about Gaussian measures on finite dimensional spaces. In anticipation of the analysisof Gaussian measures on infinite dimensional spaces, particular attention is given to thoseproperties of Gaussian measures that are dimension independent, and Gaussian processesare constructed. The rest of the book is devoted to the study of Gaussian measures onBanach spaces. The perspective adopted is the one introduced by I. Segal and developedby L. Gross in which the Hilbert structure underlying the measure is emphasized.The contents of this book should be accessible to either undergraduate or graduatestudents who are interested in probability theory and have a solid background in Lebesgueintegration theory and a familiarity with basic functional analysis. Although the focus ison Gaussian measures, the book introduces its readers to techniques and ideas that haveapplications in other contexts.

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