Domains of elliptic operators on sets in Wiener space

Abstract

Let [Formula: see text] be a separable Banach space endowed with a non-degenerate centered Gaussian measure [Formula: see text]. The associated Cameron–Martin space is denoted by [Formula: see text]. Consider two sufficiently regular convex functions [Formula: see text] and [Formula: see text]. We let [Formula: see text] and [Formula: see text]. In this paper, we study the domain of the self-adjoint operator associated with the quadratic form [Formula: see text] and we give sharp embedding results for it. In particular, we obtain a characterization of the domain of the Ornstein–Uhlenbeck operator in Hilbert space with [Formula: see text] and on half-spaces, namely if [Formula: see text] and [Formula: see text] is an affine function, then the domain of the operator defined via (0.1) is the space [Formula: see text] where [Formula: see text] is the Feyel–de La Pradelle Hausdorff–Gauss surface measure.

On functions of bounded variation on convex domains in Hilbert spaces

“…Here S(t) is the Ornstein-Uhlenbeck semigroup in (1). To conclude we prove that condition (40) is equivalent to (39).…”

“…) and the space D 1, p (X, ν), p > 1 can be defined as the domain of its closure (still denoted by D H ). In a similar way we may define D 2, p (X, ν), p ∈ (1, ∞) (for more details see [1,9,16]). The Gaussian integration by parts formula…”

“…where D H is the gradient operator along the Cameron-Martin space H (see Sect. 2) and S(t) is the classical Ornstein-Uhlenbeck semigroup defined via the Mehler formula (see (1)). The latter is the analogous, in the Gaussian setting, of the heat semigroup used by De Giorgi in [11] to provide the original definition of BV functions in the Euclidean case.…”

$L^2$-theory for transitions semigroups associated to dissipative systems

“…In Section 5 we will restrict to the case where F is a gradient perturbation, namely it has a potential. In this case the invariant measure ν is a weighted Gaussian measure and it is possible to associate a quadratic form Q 2 to N 2 , see for example [2,3,9,10,20,25,26,27,30,31,40]. Under some additional hypotheses (Hypotheses 5.5) we will define the Sobolev space W 1,2 C (X, ν), and we will show that there exists a quadratic form Q 2 on W 1,2 C (X, ν) such that…”

“…After, proceeding as in [24, Section 3], we will consider a suitable Sobolev space W 1,2 C (X, ν) of the functions u : O → R such that their null extension u belongs to W 1,2 C (X, ν), and the quadratic form…”