Surface measures in infinite-dimensional spaces

Abstract

ABSTRACT. We construct surface measures for surfaces of codimension n > 1 in Banach spaces, and in a wide class of locally convex spaces. It is assumed that the determining function has a continuous derivative along a subspace.KEY WORDS: surface measure, Radon measure, surfaces of finite codimension, Banach space, locally convex space.

Surface Measures and Tightness of (r,p)-Capacities on Poisson Space

“…In this paper, as well as in [19,20], the Sobolev smoothness conditions imposed on the functions and vector fields are weaker than those in [1]. Namely, we consider only the first-and second-order gradients of the function F determining the surface (and consider an alternative definition which involves only the first gradient of F ), and use only Sobolev capacities of the first order, hence we do not need the restriction on the manifold, formulated in condition (C).…”

“…In the case of configuration spaces, the methods of the Malliavin calculus have been employed in [6,12,25,26]. Similar techniques have been used in [8,9,[19][20][21] in the case of measures on locally convex spaces. Under more restrictive assumptions, the Gauss-Ostrogradskii formula on a configuration space has been proved in [12,26].…”

Estimate for $P_{t}D$ for the stochastic Burgers equation

“…By (15) it follows that the measure ν possesses the Fomin derivative in all directions (−A) −α h for h ∈ H, see e.g. [Pu98].…”

Capacities and Surface Measures in Locally Convex Spaces

“…Two approaches to construction of surface measures in infinite dimensional spaces are known. The first one originated in the monograph [12] by A.V.Skorohod and substantially developed in papers [14], [15], [4], [7] is based on construction of a local surface measure in quite small neighborhoods of points. A completely different approach was realized by H.Airault and P.Malliavin [1] and further developped in [2], [9], [10].…”