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 (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

\int_{E} \mathrm{D} \varphi(x) h(x) \, \mathrm{d} \mu(x) = \int_{E} \varphi(x) (A h)(x) \, \mathrm{d} \mu(x)

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(x) 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

E^{*} \xrightarrow{i^{*}} H^{*} \cong H \xrightarrow{i} E.

For h ∈ S, define Ah by

(A h)(x) = h(x) x - \mathrm{trace}_{H} \mathrm{D} h(x).

This operator A is an integration by parts operator, also known as the divergence operator; a proof can be found in Elworthy (1974).

The classical Wiener space C₀ of continuous paths in Rⁿ starting at zero and defined on the unit interval [0, 1] has another integration by parts operator. Let S be the collection

S = \left\{ \left. h \colon C_{0} \to L_{0}^{2, 1} \right| h \mbox{ is bounded and non-anticipating} \right\},

i.e., all bounded, adapted processes with absolutely continuous sample paths. Let φ : C₀ → R be any C¹ function such that both φ and Dφ are bounded. For h ∈ S and λ ∈ R, the Girsanov theorem implies that

\int_{C_{0}} \varphi (x + \lambda h(x)) \, \mathrm{d} \gamma(x) = \int_{C_{0}} \varphi(x) \exp \left( \lambda \int_{0}^{1} \dot{h}_{s} \cdot \mathrm{d} x_{s} - \frac{\lambda^{2}}{2} \int_{0}^{1} | \dot{h}_{s} |^{2} \, \mathrm{d} s \right) \, \mathrm{d} \gamma(x).

Differentiating with respect to λ and setting λ = 0 gives

\int_{C_{0}} \mathrm{D} \varphi(x) h(x) \, \mathrm{d} \gamma(x) = \int_{C_{0}} \varphi(x) (A h) (x) \, \mathrm{d} \gamma(x),

where (Ah)(x) is the Itō integral

\int_{0}^{1} \dot{h}_{s} \cdot \mathrm{d} x_{s}.

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. ISBN 0-486-44994-7. MR 2250060 (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. MR 0464297

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