Aurel I. Stan

CHARACTERIZATION OF PROBABILITY MEASURES THROUGH THE CANONICALLY ASSOCIATED INTERACTING FOCK SPACES

We continue our program of coding the whole information of a probability measure into a set of commutation relations canonically associated to it by presenting some characterization theorems for the symmetry and factorizability of a probability measure on ℝd in terms of the canonically associated interacting creation, annihilation and number operators.

A HÖLDER–YOUNG–LIEB INEQUALITY FOR NORMS OF GAUSSIAN WICK PRODUCTS

An important connection between the finite-dimensional Gaussian Wick products and Lebesgue convolution products will be proven first. Then this connection will be used to prove an important Holder inequality for the norms of Gaussian Wick products, reprove Nelson hypercontractivity inequality, and prove a more general inequality whose marginal cases are the Holder and Nelson inequalities mentioned before. We will show that there is a deep connection between the Gaussian Holder inequality and classic Holder inequality, between the Nelson hypercontractivity and classic Young inequality with the sharp constant, and between the third more general inequality and an extension by Lieb of the Young inequality with the best constant. Since the Gaussian probability measure exists even in the infinite-dimensional case, the above three inequalities can be extended, via a classic Fatous lemma argument, to the infinite-dimensional framework.

Probability measures in terms of creation, annihilation, and neutral operators

LUIGI ACCARDICentro Vito VolterraFacolt`a di EconomiaUniversit`a di Roma “Tor Vergata”00133 Roma, ItalyE-mail: accardi@volterra.mat.uniroma2.itHUI-HSIUNG KUODepartment of MathematicsLouisiana State UniversityBaton Rouge, LA 70803, USAE-mail: kuo@math.lsu.eduAUREL STANDepartment of MathematicsUniversity of RochesterRochester, NY 14627, USAE-mail: astan@math.rochester.edu

Hölder-Type Inequalities for Norms of Wick Products

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.

Some Norm Inequalities for Gaussian Wick Products

We provide several inequalities for the ℒ q (𝒫)-norm of the Wick product of random variables. These estimates are based on a Jensens type inequality for the Wick multiplication, which we derive via a positivity argument. As an application we study a certain type of anticipating stochastic differential equation whose solution is shown to be an element of ℒ q (𝒫) for some q ≥ 1.

An extension of the Beckner’s type Poincaré inequality to convolution measures on abstract Wiener spaces

ABSTRACT We generalize the Beckner’s type Poincaré inequality (Beckner, W. Proc. Amer. Math. Soc. (1989) 105:397–400) to a large class of probability measures on an abstract Wiener space of the form μ⋆ν, where μ is the reference Gaussian measure and ν is a probability measure satisfying a certain integrability condition. As the Beckner inequality interpolates between the Poincaré and logarithmic Sobolev inequalities, we utilize a family of products for functions which interpolates between the usual point-wise multiplication and the Wick product. Our approach is based on the positivity of a quadratic form involving Wick powers and integration with respect to those convolution measures. In addition, we prove that in the finite-dimensional case the class of densities of convolutions measures satisfies a point-wise covariance inequality.

A CHARACTERIZATION OF PROBABILITY MEASURES IN TERMS OF WICK PRODUCT INEQUALITIES

It is known that if X is a normally distributed random variable, and ♢ and E denote the Wick product and expectation, respectively, then for any non-negative integers m and n, and any polynomial functions f and g of degrees at most m and n, respectively, the following inequality holds: We show that this result can be extended to a random variable X, not necessary Gaussian, having an infinite support and finite moments of all orders, if and only if its Szego–Jacobi sequence {ωk}k ≥ 1 is super-additive.

BEST CONSTANTS IN NORMS OF WICK PRODUCTS

Let be a Fock space and, for any non-negative integer k, let 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 . 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.

A note on a local limit theorem for Wiener space valued random variables

We prove a local limit theorem, i.e. a central limit theorem for densities, for a sequence of independent and identically distributed random variables taking values on an abstract Wiener space; the common law of those random variables is assumed to be absolutely continuous with respect to the reference Gaussian measure. We begin by showing that the key roles of scaling operator and convolution product in this infinite dimensional Gaussian framework are played by the Ornstein-Uhlenbeck semigroup and Wick product, respectively. We proceed by establishing a necessary condition on the density of the random variables for the local limit theorem to hold true. We then reverse the implication and prove under an additional assumption the desired L1-convergence of the density of \frac{X_1+...+X_n}{\sqrt{n}}. We close the paper comparing our result with certain Berry-Esseen bounds for multidimensional central limit theorems.

A HÖLDER INEQUALITY FOR NORMS OF POISSONIAN WICK PRODUCTS

An understanding of the second quantization operator of a constant times the identity operator and the Poissonian Wick product, without using the orthogonal Charlier polynomials, is presented first. We use both understanding, with and without the Charlier polynomials, to prove some inequalities about the norms of Poissonian Wick products. These inequalities are the best ones in the case of L1, L2, and L∞ norms. We close the paper with some probabilistic interpretations of the Poissonian Wick product.