# Integration by parts operator

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Integration by parts operator

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"&lowast; are separable spaces; let "&mu;" 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" &rarr; "L"2("E", "&mu;"; R) is said to be an integration by parts operator for "&mu;" if

:$int_\left\{E\right\} mathrm\left\{D\right\} varphi\left(x\right) h\left(x\right) , mathrm\left\{d\right\} mu\left(x\right) = int_\left\{E\right\} varphi\left(x\right) \left(A h\right)\left(x\right) , mathrm\left\{d\right\} mu\left(x\right)$

for every "C"1 function "&phi;" : "E" &rarr; R and all "h" &isin; "S" for which either side of the above equality makes sense. In the above, D"&phi;"("x") denotes the Fréchet derivative of "&phi;" at "x".

Examples

* Consider an abstract Wiener space "i" : "H" &rarr; "E" with abstract Wiener measure "&gamma;". Take "S" to be the set of all "C"1 functions from "E" into "E"&lowast;; "E"&lowast; can be thought of as a subspace of "E" in view of the inclusions

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

:For "h" &isin; "S", define "Ah" by

::$\left(A h\right)\left(x\right) = h\left(x\right) x - mathrm\left\{trace\right\}_\left\{H\right\} mathrm\left\{D\right\} h\left(x\right).$

: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 paths in R"n" starting at zero and defined on the unit interval [0, 1] has another integration by parts operator. Let "S" be the collection

::$S = left\left\{ left. h colon C_\left\{0\right\} o L_\left\{0\right\}^\left\{2, 1\right\} ight| h mbox\left\{ is bounded and non-anticipating\right\} ight\right\},$

:i.e., all bounded, adapted processes with absolutely continuous sample paths. Let "&phi;" : "C"0 &rarr; R be any "C"1 function such that both "&phi;" and D"&phi;" are bounded. For "h" &isin; "S" and "&lambda;" &isin; R, the Girsanov theorem implies that

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

:Differentiating with respect to "&lambda;" and setting "&lambda;" = 0 gives

::$int_\left\{C_\left\{0 mathrm\left\{D\right\} varphi\left(x\right) h\left(x\right) , mathrm\left\{d\right\} gamma\left(x\right) = int_\left\{C_\left\{0 varphi\left(x\right) \left(A h\right) \left(x\right) , mathrm\left\{d\right\} gamma\left(x\right),$

:where ("Ah")("x") is the Itō integral

::$int_\left\{0\right\}^\left\{1\right\} dot\left\{h\right\}_\left\{s\right\} cdot mathrm\left\{d\right\} x_\left\{s\right\}.$

:The same relation holds for more general "&phi;" 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

* cite book
last = Bell
first = Denis R.
title = The Malliavin calculus
publisher = Dover Publications Inc.
location = Mineola, NY
year = 2006
pages = pp. 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&ndash;166
publisher = Internat. Atomic Energy Agency
year = 1974
MathSciNet|id=0464297

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