White Noise Integration
We define integrals of normal ordered monomials. These integrals are scalarly defined as sesquilinear forms over Open image in new window , the space of all symmetric, continuous functions of compact support with values in a Hilbert space \(\mathfrak{k}\)
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White Noise Integration
Abstract We define integrals of normal ordered monomials. These integrals are scalarly defined as sesquilinear forms over Ks (X, k), the space of all symmetric, continuous functions of compact support with values in a Hilbert space k. We can define products of those objects as scalarly defined integrals. We define C 1 -processes and calculate their Schwartz derivatives. We prove Ito’s theorem for C 1 -processes.
7.1 Integration of Normal Ordered Monomials In the following we shall, if not otherwise stated, skip Δα etc. in the integrals. So we write, e.g., μ(dα) for μ(dα)Δα. Recall that this expression stands for μ(dα) = μ(∅) +
∞ 1 · · · μ(dx1 , . . . , dxn ). n! n=1
With this simplified notation the sum-integral lemma, Theorem 2.2.1, reads
··· α1
αk
μ(dxα1 , . . . , dxαk ) =
α α +···+α =α n 1
μ(dxα1 , . . . , dxαk )
or, by neglecting the dx,
···
α1
μ(α1 , . . . , αk ) =
αk
μ(α1 , . . . , αk ).
α α +···+α =α k 1
Recall an admissible monomial is of the form (Definition 5.3.2) M = acϑnn · · · acϑ11 . W. von Waldenfels, A Measure Theoretical Approach to Quantum Stochastic Processes, Lecture Notes in Physics 878, DOI 10.1007/978-3-642-45082-2_7, © Springer-Verlag Berlin Heidelberg 2014
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White Noise Integration
Let S+ be the set of all i, such that ϑi = +1, and S− the set of all i, such that ϑi = −1. If λ is the base measure, we will use the fact, that MλS− \S+ is a positive measure in the usual sense on an appropriate space. We shall denote by Φ| the measure, concentrated on ∅, which we denoted by Ψ in Sect. 2.1 So Φ|f = f (∅). A monomial M = acϑnn · · · acϑ11 is called normal ordered if all the creators ac+ are to the left of the annihilators ac , i.e., ϑi = +1, ϑj = −1 =⇒ i > j. Using the commutation relations it is clear that any normal ordered monomial can be brought into the form a + (dxs1 ) · · · a + (dxsl )a + (dxt1 ) · · · a + (dxtm ) a(xt1 ) · · · a(xtm )a(xu1 ) · · · a(xun ) = aσ++τ aτ +υ , with σ = {s1 , . . . , sl },
τ = {t1 , . . . , tm },
υ = {u1 , . . . , un }.
Assume five finite, pairwise disjoint, index sets π, σ, τ, υ, ρ and consider the admissible monomial aπ aσ++τ aτ +υ aρ+ . The indices of creators make up the set S+ = σ +τ +ρ, and the indices of annihilators S− = π +τ +υ. So S− \S+ = π +υ. Following Sect. 5.6, aπ aσ++τ aτ +υ aρ+ λπ+υ is for fixed #π, #σ, #τ, #υ, #ρ, a measure on X #(π+σ +τ +υ+ρ) . Letting the numbers #π, #σ, #τ, #υ, #ρ run from 0 to ∞ we arrive at a measure m on X5 m = m(π, σ, τ, υ, ρ) = aπ aσ++τ aτ +υ aρ+ λπ+υ . Using Theorem 5.5.1 and Theorem 5.6.1, we obtain (forgetting about the Δω), m= aω aσ +τ aπ+ aω aτ +υ aρ+ λω+σ +τ +υ ω
=
ω
ε(σ + τ + ω, π)ε(τ + υ + ω, ρ)λω+σ +τ +υ .
If ϕ ∈ Ks (X5 ) then m(π, σ, τ, υ, ρ)ϕ(π, σ, τ, υ, ρ) =
(forgetting about the Δσ, Δτ, . . .).
ϕ(σ + τ + ω, σ, τ, υ, τ + υ + ω)λω+σ +τ +υ ,
7.1 Integration of Normal Ordered Monomials
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Assume we have a Hilbert space k with a countable basis. We often write the scalar product x, y →
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