Wick polynomials in noncommutative probability: a group-theoretical approach

Wick polynomials and Wick products are studied in the context of noncommutative probability theory. It is shown that free, Boolean, and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular Hopf algebra. These results generalize our previous developments of a Hopf-algebraic approach to cumulants and Wick products in classical probability theory.

A note on Gamma 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 sharp interpolation between the Hölder and Gaussian Young inequalities

We prove a very general sharp inequality of the Hölder–Young-type for functions defined on infinite dimensional Gaussian spaces. We begin by considering a family of commutative products for functions which interpolates between the pointwise and Wick products; this family arises naturally in the context of stochastic differential equations, through Wong–Zakai-type approximation theorems, and plays a key role in some generalizations of the Beckner-type Poincaré inequality. We then obtain a crucial integral representation for that family of products which is employed, together with a generalization of the classic Young inequality due to Lieb, to prove our main theorem. We stress that our main inequality contains as particular cases the Hölder inequality and Nelson’s hyper-contractive estimate, thus providing a unified framework for two fundamental results of the Gaussian analysis.

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.

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 Hölder inequality for the norms of Gaussian Wick products, reprove Nelson hypercontractivity inequality, and prove a more general inequality whose marginal cases are the Hölder and Nelson inequalities mentioned before. We will show that there is a deep connection between the Gaussian Hölder inequality and classic Hölder 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 Fatou's lemma argument, to the infinite-dimensional framework.

BEST CONSTANTS IN NORMS OF RETARDED WICK PRODUCTS

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.