Integration by parts operator
# 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 → $L^2(E,$ μ; R) is said to be an integration by parts operator for μ if

    $\int _{E}\mathrm {D} \varphi (x)h(x)\,\mathrm {d} \mu (x)=\int _{E}\varphi (x)(Ah)(x)\,\mathrm {d} \mu (x)$

for every $C^1$ 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 $C^1$ 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
       $(Ah)(x)=h(x)x-\mathrm {trace} _{H}\mathrm {D} h(x).$
    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 $C_0$ of continuous paths in $R^n$ 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 : $C_0$ → R be any $C^1$ 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
       $\int _{C_{0}}\mathrm {D} \varphi (x)h(x)\,\mathrm {d} \gamma (x)=\int _{C_{0}}\varphi (x)(Ah)(x)\,\mathrm {d} \gamma (x),$
    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.
