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 spaceE∗ are
separable spaces; let μ be a
Borel measure on E. Let S be any (fixed)
subset of the class of functions defined on E. A linear operator A : S → L2(E, μ; R) is said to be an integration by parts operator for μ if
for every
C1 functionφ : E → R and all h ∈ S for which either side of the above equality makes sense. In the above, Dφ(x) denotes the
Fréchet derivative of φ at x.
Examples
Consider an
abstract Wiener spacei : 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
For h ∈ S, define Ah by
This operator A is an integration by parts operator, also known as the
divergence operator; a proof can be found in Elworthy (1974).
The same relation holds for more general φ by an approximation argument; thus, the Itō integral is an integration by parts operator and can be seen as an infinite-dimensional divergence operator. This is the same result as the
integration by parts formula derived from the Clark-Ocone theorem.
References
Bell, Denis R. (2006). The Malliavin calculus. Mineola, NY: Dover Publications Inc. pp. x+113.
ISBN0-486-44994-7.
MR2250060 (See section 5.3)
Elworthy, K. David (1974). "Gaussian measures on Banach spaces and manifolds". Global analysis and its applications (Lectures, Internat. Sem. Course, Internat. Centre Theoret. Phys., Trieste, 1972), Vol. II. Vienna: Internat. Atomic Energy Agency. pp. 151–166.
MR0464297