Direct integration of a beam: Difference between revisions
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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''<sup>∗</sup> are [[separable space]]s; 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''<sup>2</sup>(''E'', ''μ''; '''R''') is said to be an '''integration by parts operator''' for ''μ'' if | |||
:<math>\int_{E} \mathrm{D} \varphi(x) h(x) \, \mathrm{d} \mu(x) = \int_{E} \varphi(x) (A h)(x) \, \mathrm{d} \mu(x)</math> | |||
for every [[smooth function|''C''<sup>1</sup> function]] ''φ'' : ''E'' → '''R''' and all ''h'' ∈ ''S'' for which either side of the above equality makes sense. In the above, D''φ''(''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''<sup>1</sup> functions from ''E'' into ''E''<sup>∗</sup>; ''E''<sup>∗</sup> can be thought of as a subspace of ''E'' in view of the inclusions | |||
::<math>E^{*} \xrightarrow{i^{*}} H^{*} \cong H \xrightarrow{i} E.</math> | |||
:For ''h'' ∈ ''S'', define ''Ah'' by | |||
::<math>(A h)(x) = h(x) x - \mathrm{trace}_{H} \mathrm{D} h(x).</math> | |||
: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''<sub>0</sub> of [[continuous function|continuous paths]] in '''R'''<sup>''n''</sup> starting at zero and defined on the [[interval (mathematics)|unit interval]] [0, 1] has another integration by parts operator. Let ''S'' be the collection | |||
::<math>S = \left\{ \left. h \colon C_{0} \to L_{0}^{2, 1} \right| h \mbox{ is bounded and non-anticipating} \right\},</math> | |||
:i.e., all [[bounded function|bounded]], [[adapted process|adapted]] processes with [[absolutely continuous]] sample paths. Let ''φ'' : ''C''<sub>0</sub> → '''R''' be any ''C''<sup>1</sup> function such that both ''φ'' and D''φ'' are bounded. For ''h'' ∈ ''S'' and ''λ'' ∈ '''R''', the [[Girsanov theorem]] implies that | |||
::<math>\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).</math> | |||
:Differentiating with respect to ''λ'' and setting ''λ'' = 0 gives | |||
::<math>\int_{C_{0}} \mathrm{D} \varphi(x) h(x) \, \mathrm{d} \gamma(x) = \int_{C_{0}} \varphi(x) (A h) (x) \, \mathrm{d} \gamma(x),</math> | |||
:where (''Ah'')(''x'') is the [[Itō integral]] | |||
::<math>\int_{0}^{1} \dot{h}_{s} \cdot \mathrm{d} x_{s}.</math> | |||
: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 [[Clark-Ocone theorem#Integration by parts on Wiener space|integration by parts formula derived from the Clark-Ocone theorem]]. | |||
==References== | |||
* {{cite book | |||
| last = Bell | |||
| first = Denis R. | |||
| title = The Malliavin calculus | |||
| publisher = Dover Publications Inc. | |||
| location = Mineola, NY | |||
| year = 2006 | |||
| pages = x+113 | |||
| isbn = 0-486-44994-7 | |||
}} {{MathSciNet|id=2250060}} (See section 5.3) | |||
* {{cite book | |||
| last = Elworthy | |||
| first = K. David | |||
| chapter = Gaussian measures on Banach spaces and manifolds | |||
| title = Global analysis and its applications (Lectures, Internat. Sem. Course, Internat. Centre Theoret. Phys., Trieste, 1972), Vol. II | |||
| pages = 151–166 | |||
| publisher = Internat. Atomic Energy Agency | |||
| address = Vienna | |||
| year = 1974 | |||
}} {{MathSciNet|id=0464297}} | |||
[[Category:Integral calculus]] | |||
[[Category:Measure theory]] | |||
[[Category:Operator theory]] | |||
[[Category:Stochastic processes]] |
Latest revision as of 02:38, 2 February 2014
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
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
- For h ∈ S, define Ah by
- 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
- 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
- Differentiating with respect to λ and setting λ = 0 gives
- where (Ah)(x) is the Itō integral
- 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.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Template:MathSciNet (See section 5.3) - 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