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 &mu; 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, &mu;; R) is said to be an integration by parts operator for &mu; 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 &phi; : E → R and all h ∈ S for which either side of the above equality makes sense. In the above, D&phi;(x) denotes the Fréchet derivative of &phi; at x.

Examples

 * Consider an abstract Wiener space i : H → E with abstract Wiener measure &gamma;. 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 &isin; 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 &phi; : C0 &rarr; R be any C1 function such that both &phi; and D&phi; are bounded.  For h &isin; S and &lambda; &isin; 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 &lambda; and setting &lambda; = 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 &phi; 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.