Integration by parts operator

In mathematics, an integration by parts operator is a linear operator used to formulate integration by parts formulae; the most interesting examples of integration by parts operators occur in infinite-dimensional settings and find uses in stochastic analysis and its applications.

Definition

Let E be a Banach space such that both E and its continuous dual space E^∗ are separable spaces; let μ be a Borel measure on E. Let S be any subset of the class of functions defined on E. A linear operator A : S → L² is said to be an integration by parts operator for μ if
for every C¹ function φ : E → R and all h ∈ S for which either side of the above equality makes sense. In the above, Dφ denotes the Fréchet derivative of φ at x.

Examples

Consider an abstract Wiener space i : H → E with abstract Wiener measure γ. Take S to be the set of all C¹ functions from E into E^∗; E^∗ can be thought of as a subspace of E in view of the inclusions
The classical Wiener space C₀ of continuous paths in Rⁿ starting at zero and defined on the unit interval has another integration by parts operator. Let S be the collection