The Malliavin calculus, a functional analytic approach

Abstract

In his groundbreaking articles [S, 91, Malliavin introduced a technique for obtaining elliptic regularity results using function space calculus. In his formulation, the function space calculus on which his theory rests is the martingale calculus coming from the Ornstein-Uhlenbeck process on Wiener space (cf. [ 121 for a detailed discussion of Malliavin’s theory from this point of view). Although this formulation has a great deal to recommend it and is particularly pleasing to afficionados of the so-called “Brownian sheet,” most analysts (including those who are reasonably facile with the machinery of probability theory) are unlikely to find the technical difficulties inherent in this approach outweighed by the eventual rewards. Shigekawa [ 111 discovered that there is an alternative formulation of Malliavin’s theory. In Shigekawa’s formulation, no mention need be made of the Ornstein-Uhlenbeck process. Instead, what he relies on is a Sobolev type extension of the Frechet dervative with Wiener measure playing the role played by Lebesgue measure in the finite dimensional context. A third formulation of Malliavin’s theory was recently provided by Bismut [ l]. Bismut’s idea is to exploit the quasi-invariance of Wiener measure as it is manifested in a beautiful relation due to Haussmann [4]. The present article has two aims. In the first place, Section 1, 2, and 3 are devoted to yet another formulation of Malliavin’s calculus. Although the end results are precisely the same as those obtained in [ 121, no reference is made to the Ornstein-Uhlenbeck process on which the original theory depended. Instead, the machinery is entirely that of elementary functional analysis and is devoid of any sophisticated martingale calculus. It has been my intent that these three sections should be readily accessible to any analyst who has had some acquintance with Wiener measure.