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
∫ E D φ ( x ) h ( x ) d μ ( x ) = ∫ E φ ( x ) ( A h ) ( x ) d μ ( x ) {\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
E ∗ → i ∗ H ∗ ≅ H → i E . {\displaystyle E^{*}{\xrightarrow {i^{*}}}H^{*}\cong H{\xrightarrow {i}}E.}
For h ∈ S, define Ah by
( A h ) ( x ) = h ( x ) x − t r a c e H D h ( x ) . {\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
S = { h : C 0 → L 0 2 , 1 | h is bounded and non-anticipating } , {\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
∫ C 0 φ ( x + λ h ( x ) ) d γ ( x ) = ∫ C 0 φ ( x ) exp ( λ ∫ 0 1 h ˙ s ⋅ d x s − λ 2 2 ∫ 0 1 | h ˙ s | 2 d s ) d γ ( x ) . {\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
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