The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Gaussian Measures on a Banach Space As I said at the end of § 4.3.2, the distribution of Brownian motion is called Wiener measure because Wiener was the first to construct it. Wiener's own thinking about his measure had little or nothing in common with the Lévy-Khinchine program. Instead, he looked upon his measure as a Gaussian measure on an infinite dimensional space, and most of what he did with his measure is best understood from that perspective. Thus, in this chapter, we will look at Wiener measure from a strictly Gaussian point of view. More generally, we will be dealing here with measures on a real Banach space E which are centered Gaussian in the sense that, for each x * in the dual space E * , x ∈ E −→ x, x * ∈ R is a centered Gaussian random variable. Not surprisingly, such a measure will be said to be a centered Gaussian measure on E. Although the ideas which I will use are implicit in Wiener's work, it was I. Segal and his school, especially L. Gross, * who gave them the form presented here. § 8.1 The Classical Wiener Space In order to motivate what follows, it is helpful to first understand Wiener measure from the point of view which I will be adopting here.

Mathematics Subject Classification. Primary 35-02, 35J15, 35K10, 60J35, 60J60. For additional information and updates on this book, visit www.ams.org/bookpages/surv-207 Library of Congress Cataloging-in-Publication Data Fokker-Planck-Kolmogorov equations /Vladimir I. Bogachev, Nicolai V. Krylov, Michael Röckner, Stanislav V. Shaposhnikov. pages cm. -(Mathematical surveys and monographs ; volume 207) Includes bibliographical references and index. ISBN 978-1-4704-2558-6 (alk. paper) 1. Fokker-Planck equation. 2. Stochastic differential equations. I. Bogachev, V. I. (Vladimir Igorevich), 1961-II. Krylov, N. V. (Nicolai Vladimirovich). III. Röckner, Michael, 1956-IV. Shaposhnikov, Stanislav V. QA274.23.F65 2015 515 .353-dc23 2015024922Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

No abstract

No abstract