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