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.


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".


* 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.


* 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

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Self-adjoint operator — In mathematics, on a finite dimensional inner product space, a self adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose.… …   Wikipedia

  • Dissipative operator — In mathematics, a dissipative operator is a linear operator A defined on a linear subspace D(A) of Banach space X, taking values in X such that for all λ > 0 and all x ∈ D(A) A dissipative operator is called maximally… …   Wikipedia

  • Laplace operator — This article is about the mathematical operator. For the Laplace probability distribution, see Laplace distribution. For graph theoretical notion, see Laplacian matrix. Del Squared redirects here. For other uses, see Del Squared (disambiguation) …   Wikipedia

  • Summation by parts — In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. The summation by parts formula is sometimes called… …   Wikipedia

  • Ornstein–Uhlenbeck operator — Not to be confused with Ornstein–Uhlenbeck process. In mathematics, the Ornstein–Uhlenbeck operator can be thought of as a generalization of the Laplace operator to an infinite dimensional setting. The Ornstein–Uhlenbeck operator plays a… …   Wikipedia

  • Mobile virtual network operator — A mobile virtual network operator (MVNO) is a company that provides mobile phone services but does not have its own licensed frequency allocation of radio spectrum, nor does it necessarily have all of the infrastructure required to provide mobile …   Wikipedia

  • List of mathematics articles (I) — NOTOC Ia IA automorphism ICER Icosagon Icosahedral 120 cell Icosahedral prism Icosahedral symmetry Icosahedron Icosian Calculus Icosian game Icosidodecadodecahedron Icosidodecahedron Icositetrachoric honeycomb Icositruncated dodecadodecahedron… …   Wikipedia

  • Clark–Ocone theorem — In mathematics, the Clark–Ocone theorem (also known as the Clark–Ocone–Haussmann theorem or formula) is a theorem of stochastic analysis. It expresses the value of some function F defined on the classical Wiener space of continuous paths starting …   Wikipedia

  • Clark-Ocone theorem — In mathematics, the Clark Ocone theorem (also known as the Clark Ocone Haussmann theorem or formula) is a theorem of stochastic analysis. It expresses the value of some function F defined on the classical Wiener space of continuous paths starting …   Wikipedia

  • Dirac delta function — Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually used to specify the value of any multiplicative constant, which will give the area under the function. The other convention… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.