Direct integration of a beam

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

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

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

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

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 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 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

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

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.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Template:MathSciNet