On integration by parts formula on open convex sets in Wiener spaces

Abstract

In Euclidean space, it is well known that any integration by parts formula for a set of finite perimeter Ω is expressed by the integration with respect to a measure P (Ω, •) which is equivalent to the one-codimensional Hausdorff measure restricted to the reduced boundary of Ω. The same result has been proved in an abstract Wiener space, typically an infinite dimensional space, where the surface measure considered is the one-codimensional spherical Hausdorff-Gauss measure S ∞−1 restricted to the measure-theoretic boundary of Ω. In this paper we consider an open convex set Ω and we provide an explicit formula for the density of P (Ω, •) with respect to S ∞−1 . In particular, the density can be written in terms of the Minkowski functional p of Ω with respect to an inner point of Ω. As a consequence, we obtain an integration by parts formula for open convex sets in Wiener spaces.

Surface measures and integration by parts formula on levels sets induced by functionals of the Brownian motion in $${\mathbb {R}}^n$$

“…The integration by parts formula. In this section we discuss the integration by parts formula on the level sets of the mapping g. Similar results have been obtained by [8,1] with different techniques, see also [6,Section 4].…”

“…The aim of this paper is to construct the surface measure induced by µ on the level sets {g = r} and provide an integration by parts formula involving this surface measure. We shall mention here that, since the domain {g < r} is a convex open set in E, our construction is related to that of the recent paper [1]. In particular, the integration by parts formula that we obtain in Proposition 4.8 is related to formula (1) in [1].…”

“…We shall mention here that, since the domain {g < r} is a convex open set in E, our construction is related to that of the recent paper [1]. In particular, the integration by parts formula that we obtain in Proposition 4.8 is related to formula (1) in [1]. Notice however that our construction is quite different.…”

“…Let E = C([0, 1]; R n ) denote the Banach space of continuous functions from [0, 1] to R n , endowed with the sup-norm f ∞ = sup [0,1] |f (x)|. We denote by E = B(E) the σ-field of Borel measurable subsets of E. Also, we introduce the Hilbert space H = L 2 (0, 1; R n ) of square integrable measurable functions.…”

“…In case u = B, we shall simply write g(x) = g(B)(x) = 1 2 x 2 H . The aim of this paper is to construct the surface measure induced by µ on the level sets {g = r} and provide an integration by parts formula involving this surface measure.…”