Integration by parts operator


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"∗ 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" → "L"2("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) (A h)(x) , mathrm{d} mu(x)

for every "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

::E^{*} xrightarrow{i^{* H^{*} cong H xrightarrow{i} E.

:For "h" ∈ "S", define "Ah" by

::(A h)(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 "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. h colon C_{0} o L_{0}^{2, 1} ight| h mbox{ is bounded and non-anticipating} ight},

:i.e., all bounded, 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

::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 ight) , 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) (A h) (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

* 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 = 151–166
publisher = Internat. Atomic Energy Agency
address = Vienna
year = 1974
MathSciNet|id=0464297


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