Analysis on the Wiener Space

Abstract

In this chapter we consider the particular case where the normal martin- gale (Mt)t?R+ is a standard Brownian motion. The general results stated in Chapters 3 and 4 are developed in this particular setting of a continuous martingale. Here, the gradient operator has the derivation property and can be interpreted as a derivative in the directions of Brownian paths, while the multiple stochastic integrals are connected to the Hermite polynomials. The connection is also made between the gradient and divergence operators and other transformations of Brownian motion, e.g. by time changes. We also de- scribe in more detail the specific forms of covariance identities and deviation inequalities that can be obtained on the Wiener space and on Riemannian path space.