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

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

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

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

For h ∈ S, define Ah by

${\displaystyle (Ah)(x)=h(x)x-\mathrm {trace} _{H}\mathrm {D} h(x).}$

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 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

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

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

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

Differentiating with respect to λ and setting λ = 0 gives

${\displaystyle \int _{C_{0}}\mathrm {D} \varphi (x)h(x)\,\mathrm {d} \gamma (x)=\int _{C_{0}}\varphi (x)(Ah)(x)\,\mathrm {d} \gamma (x),}$

where (Ah)(x) is the Itō integral

${\displaystyle \int _{0}^{1}{\dot {h}}_{s}\cdot \mathrm {d} x_{s}.}$

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