Direct integration of a beam

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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(EμR) is said to be an integration by parts operator for μ if

EDφ(x)h(x)dμ(x)=Eφ(x)(Ah)(x)dμ(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*HiE.
For h ∈ S, define Ah by
(Ah)(x)=h(x)xtraceHDh(x).
This operator A is an integration by parts operator, also known as the divergence operator; a proof can be found in Elworthy (1974).
S={h:C0L02,1|h is bounded and non-anticipating},
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
C0φ(x+λh(x))dγ(x)=C0φ(x)exp(λ01h˙sdxsλ2201|h˙s|2ds)dγ(x).
Differentiating with respect to λ and setting λ = 0 gives
C0Dφ(x)h(x)dγ(x)=C0φ(x)(Ah)(x)dγ(x),
where (Ah)(x) is the Itō integral
01h˙sdxs.
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