Calculus on Gaussian and Poisson white noises

Ito, Yoshifusa; Kubo, Izumi

doi:10.1017/s0027763000000994

Cited by 72 publications

(55 citation statements)

References 12 publications

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“…(see, e.g., [10]). It follows from (2.1) that the Schwartz space S(R d ) is not only a nuclear space, but also a nuclear algebra.…”

Section: Construction Of the Probability Spacementioning

confidence: 99%

Infinite dimensional analysis of pure jump Lévy processes on the Poisson space

Løkka¹,

Proske²

2006

MATH. SCAND.

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Abstract. We develop a white noise calculus for pure jump Lévy processes on the Poisson space. This theory covers the treatment of Lévy processes of unbounded variation. The starting point of the theory is the construction of a distribution space. This space has many of the same nice properties as the classical Schwartz space, but is modified in a certain way in order to be more suitable for pure jump Lévy processes. We apply Minlos's theorem to this space and obtain a white noise measure which satisfies the first condition of analyticity, and which is non-degenerate. Furthermore, we obtain generalized Charlier polynomials for all Lévy measures. We introduce Kondratiev test function and distribution spaces, the S-transform and the Wick product. We proceed by using a transfer principle on Poisson spaces to establish a differential calculus.

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“…(see, e.g., [10]). It follows from (2.1) that the Schwartz space S(R d ) is not only a nuclear space, but also a nuclear algebra.…”

Section: Construction Of the Probability Spacementioning

confidence: 99%

Infinite dimensional analysis of pure jump Lévy processes on the Poisson space

Løkka¹,

Proske²

2006

MATH. SCAND.

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“…One can see the correspondence between classical and quantum white noises of Gaussian and Poisson types. In the following, we quickly summarize the essence of Gaussian white noise theory from [19][20] [23] and Poisson white noise theory from [13]. From now on, we always suppose (H1)(H2)(H3) to discuss the Gaussian part.…”

Section: Gel'fand Triples In Terms Of Multiple Wiener-itô Integrals Amentioning

confidence: 99%

“…Before stating and proving this theorem, let us point out that the multiplication operator byḂ(t) has the expression [19], (4.1)Ḃ(t)· = ∂ t,G + ∂ * t,G , and the multiplication operator byṖ (t) has the form [13], (4.2)Ṗ (t)· = ∂ t,P + ∂ * t,P + ∂ * t,P ∂ t,P + I. Remember that ∂ t,X and ∂ * t,X are the operators in the stage of Schrödinger representation.…”

Section: Characterization Theorems and Quantum White Noisesmentioning

confidence: 99%

“…Parts of the results in this paper were based upon Asai's talk in the AMS special session "Analysis on Infinite Dimensional Spaces (in honor of L. Gross)" during the AMS-MAA Joint Mathematics Meetings in New Orleans, USA, January [10][11][12][13] 2001. He wants to give his deepest appreciation to Professors H.-H. Kuo and A. Sengupta for the warm hospitality during his visit to Louisiana State University (January [8][9][10][11][12][13][14][15][16] 2001).…”

Section: Acknowledgmentsmentioning

confidence: 99%

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Gaussian and Poisson white noises with related characterization theorems

Asai¹,

Kubo²,

Kuo³

2003

Finite and Infinite Dimensional Analysis in Honor of Leonard Gross

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Dedicated to Professor Leonard Gross on the occasion of his 70th birthday.Abstract. Let µ G and µ P be a Gaussian measure and a Poisson measure on E * , respectively. Let at and a * t be respectively annihilation and creation operators at a point t ∈ R. In the theory of quantum white noise, it is known that at is a continuous linear operator from Γu(E C ) into itself and a * t is a continuous linear operator from Γu(E C ) * into itself. In paticular, at + a * t and at+a * t +a * t at+I are called the quantum Gaussian white noise and the quantum Poisson white noise, respectively. The main purpose of this work is to realize quantum Gaussian and Poisson white noises in terms of multiple Wiener-Itô integrals, and show that such realizations cannot be achieved by J-transform and its holomorphy, but can be done by S X -transform depending on the exponential function φ X ξ , which determines a unitary isomorphism between Boson Fock space and L 2 (E * , µ X ), X = G, P . In Appendix A, some connections between [6][7] and [9] will be discussed.

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“…Then it is a continuous (non-linear) functional of x e $* because of Lemma 2.1 and of the following estimation: The left hand side is equal to the Hilbert-Schmidt operator norm of the injection c {m>^p) by the proof of Proposition 3.6 in [9].…”

Section: *=O (N -2k)\k\mentioning

confidence: 99%

A remark on the space of testing random variables in the white noise calculus

Kubo

Yokoi

1989

Nagoya Mathematical Journal

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The first author and S. Takenaka introduced the structure of a Gel’fand triplet ℋ ⊂ (L2) ⊂ ℋ* into Hida’s calculus on generalized Brownian functionals [4-7]. They showed that the space ℋ of testing random variables has nice properties. For example, ℋ is closed under multiplication of two elements in ℋ, each element of ℋ is a continuous functional on the basic space ℰ*, in addition it can be considered as an analytic functional, and moreover exp [tΔv] (Δv is Volterra’s Laplacian) is real analytic in t ∊ R as a one-parameter group of operators on ℋ, etc.

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Calculus on Gaussian and Poisson white noises

Cited by 72 publications

References 12 publications

Infinite dimensional analysis of pure jump Lévy processes on the Poisson space

Infinite dimensional analysis of pure jump Lévy processes on the Poisson space

Gaussian and Poisson white noises with related characterization theorems

A remark on the space of testing random variables in the white noise calculus

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