On the Space of BV Functions and a Related Stochastic Calculus in Infinite Dimensions

Abstract

dedicated to professor leonard gross on the occasion of his 70th birthdayFunctions of bounded variation (BV functions) are defined on an abstract Wiener space (E, H, +) in a way similar to that in finite dimensions. Some characterizations are given, which justify describing a BV function as a function in L(log L)1Â2 with the first order derivative being an H-valued measure. It is also shown that the space of BV functions is obtained by a natural extension of the Sobolev space D 1, 1 . Moreover, some stochastic formulae related to BV functions are investigated. Academic Press

Some Fine Properties of BV Functions on Wiener Spaces

“…The theory started with the papers [14], [15] where essentially a probabilistic approach was given and has been subsequently developed in [17], [5] with a more analytic approach.…”

Mosco convergence of Dirichlet forms in infinite dimensions with changing reference measures

“…Under the condition 1 Recall that a function f is called BV if the weak derivative of f is a measure of bounded variation (see [13]). Some results on the BV functions from the point of view of the Dirichlet forms theory are obtained in [14,15]. Such examples can be easily generalized to higher dimensions.…”

Integration by parts formulae for Wiener measures restricted to subsets in Rd

“…In the setting of an abstract Wiener space, several description of Caccioppoli sets are given in detail in [2,3]. IbP formulae (in fact, divergence formulae) in infinite dimensions have also been studied by Goodman [5], Shigekawa [11] and other researchers in various contexts.…”