BEST CONSTANTS IN NORMS OF WICK PRODUCTS

2006 ◽  
Vol 09 (02) ◽  
pp. 169-185 ◽  
Author(s):  
AUREL I. STAN
Keyword(s):  
Lower Bound ◽  
Fock Space ◽  
Best Constants ◽  
Negative Integer ◽  
Wick Product ◽  
Bilinear Operator ◽  
Wick Products

Let [Formula: see text] be a Fock space and, for any non-negative integer k, let [Formula: see text] be the sum of all homogeneous chaos spaces of order at most k. For all non-negative integers m and n, the Wick product is a bounded bilinear operator from Γ (Hc)m × Γ (Hc)n into Γ (Hc)m +n with norm greater than or equal to [Formula: see text]. In the monograph1 S. Janson conjectured that this lower bound is exact. In this paper we prove this conjecture. In addition, we prove that a pair of nonzero vectors (ϕ, ψ)∈ Γ (Hc)m × Γ (Hc)n achieve this bound if and only if both vectors are multiples of the homogeneous products, i.e. ϕ =αu⊗m, ψ =βu⊗n, with u, v∈ Hc and α, β ∈ ℂ.

BEST CONSTANTS IN NORMS OF RETARDED WICK PRODUCTS

2010 ◽  
Vol 13 (03) ◽  
pp. 347-361 ◽  
Author(s):  
AUREL I. STAN
Keyword(s):  
Random Vector ◽  
Best Constants ◽  
Wick Product ◽  
Best Constant ◽  
Wick Products ◽  
Normally Distributed

We find the best constant c(m, n, r), such that the inequality: [Formula: see text] holds for all polynomials f and g of degree at most m and n, respectively, where X is a normally distributed random vector, ⋄r denotes the r-retarded Wick product and ‖⋅‖ the L2-norm.

scholarly journals Hölder-Type Inequalities for Norms of Wick Products

2008 ◽  
Vol 2008 ◽  
pp. 1-22 ◽  
Author(s):  
Alberto Lanconelli ◽  
Aurel I. Stan
Keyword(s):  
Random Variable ◽  
Upper Bounds ◽  
Minimal Condition ◽  
Wick Product ◽  
Identity Operator ◽  
Second Quantization ◽  
Measurable Functions ◽  
Quantization Operator ◽  
Wick Products ◽  
Finite Moments

Various upper bounds for the L2-norm of the Wick product of two measurable functions of a random variable X, having finite moments of any order, together with a universal minimal condition, are proven. The inequalities involve the second quantization operator of a constant times the identity operator. Some conditions ensuring that the constants involved in the second quantization operators are optimal, and interesting examples satisfying these conditions are also included.

A note on Gamma Wick products

2018 ◽  
Vol 21 (01) ◽  
pp. 1850004 ◽  
Author(s):  
Florin Catrina ◽  
Aurel I. Stan
Keyword(s):  
Integral Representation ◽  
Random Variables ◽  
Hölder Inequality ◽  
Wick Product ◽  
Wick Products

An integral representation of the Wick product for Gamma distributed random variables, with mean greater than [Formula: see text], is presented first. We use this integral representation to prove a Hölder inequality for norms of Gamma Wick products.

A NOTE ON CONVOLUTION OPERATORS IN WHITE NOISE CALCULUS

2011 ◽  
Vol 14 (04) ◽  
pp. 661-674 ◽  
Author(s):  
NOBUAKI OBATA ◽  
HABIB OUERDIANE
Keyword(s):  
Laplace Transform ◽  
White Noise ◽  
Convolution Operator ◽  
Fock Space ◽  
Wick Product ◽  
Convolution Product ◽  
Convolution Operators ◽  
S Transform ◽  
The Laplace Transform ◽  
White Noise Calculus

We derive some characteristic properties of the convolution operator acting on white noise functions and prove that the convolution product of white noise distributions coincides with their Wick product. Moreover, we show that the S-transform and the Laplace transform coincide on Fock space.

The Fröhlich Hamiltonian: mathematical results. I. Hamiltonian with cut-off

1976 ◽  
Vol 54 (15) ◽  
pp. 1613-1620 ◽  
Author(s):  
Bernard M. de Dormale ◽  
Georges Bader
Keyword(s):  
Lower Bound ◽  
Coulomb Interaction ◽  
Fock Space ◽  
Perturbation Series ◽  
Operator Norm

We study the Fröhlich Hamiltonian with cut-off. We show that this Hamiltonian is self-adjoint in the Fock space and semibounded when the number of electrons is kept constant. The lower bound we give generalizes the bound obtained for one electron by Lee and Pines to the case of several electrons. We also give a domain where the perturbation series for the resolvent of the Hamiltonian does converge in the operator norm. Finally, we study the influence of the Coulomb interaction.

Stochastic Heat Transfer in Fins and Transient Cooling Using Polynomial Chaos and Wick Products

2006 ◽  
Vol 129 (9) ◽  
pp. 1127-1133 ◽  
Author(s):  
A. F. Emery ◽  
D. Bardot
Keyword(s):  
Heat Transfer ◽  
White Noise ◽  
Polynomial Chaos ◽  
Wick Product ◽  
Perturbation Approach ◽  
Strongly Correlated ◽  
Mean Values ◽  
Wick Products ◽  
Product Approach ◽  
Transfer Problems

Stochastic heat transfer problems are often solved using a perturbation approach that yields estimates of mean values and standard deviations for properties and boundary conditions that are random variables. Methods based on polynomial chaos and Wick products can be used when the randomness is a random field or white noise to describe specific realizations and to determine the statistics of the response. Polynomial chaos is best suited for problems in which the properties are strongly correlated, while the Wick product approach is most effective for variables containing white noise components. A transient lumped capacitance cooling problem and a one-dimensional fin are analyzed by both methods to demonstrate their usefulness.

scholarly journals Compositions of states and observables in Fock spaces

2019 ◽  
Vol 32 (05) ◽  
pp. 2050012
Author(s):  
L. Amour ◽  
L. Jager ◽  
J. Nourrigat
Keyword(s):  
Quantum Electrodynamics ◽  
Fock Space ◽  
The Other ◽  
Wick Product ◽  
Fock Spaces ◽  
Weyl Symbol ◽  
Time Dynamics ◽  
Bargmann Transforms ◽  
Identification Operator

This article is concerned with compositions in the context of three standard quantizations in the framework of Fock spaces, namely, anti-Wick, Wick and Weyl quantizations. The first one is a composition of states also known as a Wick product and is closely related to the standard scattering identification operator encountered in Quantum Electrodynamics for issues on time dynamics (see [ 29 , 13 ]). Anti-Wick quantization and Segal–Bargmann transforms are implied here for that purpose. The other compositions are for observables (operators in some specific classes) for the Wick and Weyl symbols. For the Wick and Weyl symbols of the composition of two operators, we obtain an absolutely converging series and for the Weyl symbol, the remainder terms up to any orders of the expansion are controlled, still in the Fock space framework.

γ-product of white noise space and applications

2017 ◽  
Vol 25 (4) ◽  
Author(s):  
Samah Horrigue
Keyword(s):  
Differential Equations ◽  
White Noise ◽  
Ordinary Differential Equations ◽  
Existence And Uniqueness ◽  
Fock Space ◽  
Wick Product ◽  
Uniqueness Of Solutions ◽  
White Noise Space

AbstractIn this paper, we define and give some characteristic properties of γ-product in white noise space, which is the generalization of the Wick product. Existence and uniqueness of solutions are proved for a certain class of ordinary differential equations for the Fock space.

Pascal white noise harmonic analysis on configuration spaces and applications

2018 ◽  
Vol 21 (04) ◽  
pp. 1850024
Author(s):  
Anis Riahi ◽  
Habib Rebei
Keyword(s):  
White Noise ◽  
Harmonic Analysis ◽  
Configuration Space ◽  
Fock Space ◽  
Noise Analysis ◽  
Random Measure ◽  
Configuration Spaces ◽  
Wick Product ◽  
White Noise Analysis ◽  
Natural Realization

In this paper, we unify techniques of Pascal white noise analysis and harmonic analysis on configuration spaces establishing relations between the main structures of both ones. Fix a Random measure [Formula: see text] on a Riemannian manifold [Formula: see text], we construct on the space of finite compound configuration space [Formula: see text] the so-called Lebesgue–Pascal measure [Formula: see text] and as a consequence we obtain the Pascal measure [Formula: see text] on the compound configuration space [Formula: see text]. Next, the natural realization of the symmetric Fock space over [Formula: see text] as the space [Formula: see text] leads to the unitary isomorphism [Formula: see text] between the space [Formula: see text] and [Formula: see text]. Finally, in the first application we study some algebraic products, namely, the Borchers product on the Fock space, the Wick product on the Pascal space, and the ⋆-convolution on the Lebesgue–Pascal space and we prove that the Pascal white noise analysis and harmonic analysis are related through an equality of operators involving [Formula: see text]. The second application is devoted to solve the implementation problem.

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