Fock Space and the Poisson Process

Pathmanathan, Sureka; Vincent-Smith, G. F.

doi:10.1112/s0024611504014820

Cited by 1 publication

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“…We work with exponential vectors and Wick exponentials rather than finite tensor products and Wick polynomials. A parallel approach to generalized Poisson processes, based on an earlier version of this article, may be found in [30,31].…”

Section: Introductionmentioning

confidence: 99%

“…We are particularly interested in the extent to which classical commutative properties lie within and derive from non-commutative models. For example, the correction term in the classical and quantum Itô's formula is a direct consequence of the following form of the infinitesimal commutation relations of Hudson and Parthasarathy [11,27,31,41]: It is well known that W and S are identical, however, we distinguish them to emphasize the difference in their construction [5,15,23,36].…”

Section: Introductionmentioning

confidence: 99%

“…Further evidence of the value of a C*-algebraic approach to such problems may be found in [30,31]. A commutative C*-algebra generated by projective unitary representations of the group of rigid motions of H is used in place of A (see [27, IV, § 1.9]) to intrinsically construct a generalized Poisson processes I.…”

Section: Introductionmentioning

confidence: 99%

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C*-algebras and the Malliavin calculus

Vincent-Smith

2010

Proceedings of the London Mathematical Society

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We develop an intrinsic C*-algebraic model of a Gaussian field over a Hilbert space H. The model contains Gross' abstract Wiener spaces, Malliavin's Gaussian probability spaces and Itô's Wiener space. Within this model we exhibit correspondences between algebraic properties of symmetric Fock space over H and analytic properties of Malliavin calculus. As a consequence we obtain, prima facie, a canonical setting for Malliavin calculus. Mathematics Subject Classification 60G15, 60H07 (primary), 46J25, 46L53 (secondary).Proof. Linear independence follows from a theorem of Artin [2, Theorem 12]. If χ h (h 1 ) = χ h (h 2 ) for all h ∈ H r , then e it(h 1 −h 2 )(h) = 1 for all h ∈ H r and real t. Differentiating gives (h 1 − h 2 )(h) = 0 for all h ∈ H r and h 1 = h 2 .Definition 2.2 (Almost periodic compactification). The C*-algebra AP = AP H of almost periodic functions on H r is the closure in C b (H r ) of the *-subalgebraIn short,The almost periodic compactification of H r is the spectrum Ω ≡ Ω H of AP [10], [18, op. cit.].The underlying measurable spaces of our Gaussian fields will be (Ω, B) ≡ (Ω H , B H ) where B is the Baire σ-field of Ω. Suffixes are dropped when the meaning is clear.

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Section: Introductionmentioning

confidence: 99%