Integration by parts operator

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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[edit]

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 → L2(EμR) 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 C1 function φ : E → R and all h ∈ S for which either side of the above equality makes sense. In the above, Dφ(x) denotes the Fréchet derivative of φ at x.

Examples[edit]

  • Consider an abstract Wiener space i : H → E with abstract Wiener measure γ. Take S to be the set of all C1 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).
S = \left\{ \left. h \colon C_{0} \to L_{0}^{2, 1} \right| h \mbox{ is bounded and non-anticipating} \right\},
i.e., all bounded, adapted processes with absolutely continuous sample paths. Let φ : C0 → R be any C1 function such that both φ and Dφ 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 \right) \, \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[edit]

  • Bell, Denis R. (2006). The Malliavin calculus. Mineola, NY: Dover Publications Inc. pp. x+113. ISBN 0-486-44994-7.  MR 2250060 (See section 5.3)
  • Elworthy, K. David (1974). "Gaussian measures on Banach spaces and manifolds". Global analysis and its applications (Lectures, Internat. Sem. Course, Internat. Centre Theoret. Phys., Trieste, 1972), Vol. II. Vienna: Internat. Atomic Energy Agency. pp. 151–166.  MR 0464297