Direct integration of a beam

2014-02-02T00:38:32Z

71.90.103.119: /* Sample calculations */

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 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''2(''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''1 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''1 functions from ''E'' into ''E''∗; ''E''∗ 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''0 of [[continuous function|continuous paths]] in '''R'''''n'' 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''0 → '''R''' be any ''C''1 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]]

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Direct integration of a beam