Gaussian Sobolev Spaces and Stochastic Calculus of Variations

Abstract

In Chapter IV, we began by basing probability theory on the theory of abstract measure spaces of Chapter I. We then studied convergence in distribution by means of the Fourier transform on R^d . Thus both abstract integration theory and classical analysis were necessary to obtain the limit theorems of probability theory. These two sources of Chapter IV derive from the dual nature of distributions. Although a distribution is attached to a very abstract object, a random variable on a probability space, it can also be thought of as given by a Radon measure on R. Borrowing an image from Plato, we might say that distributions have a daemonic nature: they come simultaneously from celestial objects (the abstract theory of measure spaces) and terrestrial objects (analysis on R).