arXiv subjects
Publications and source records attributed to Aurel I. Stan.
We find the position-momentum decomposition of the quantum operators of the classic Meixner random variables. The position-momentum decomposition involves translation operators, which are used to give a new characterization of the Meixner random variables.
A definition of $d$--dimensional $n$--Meixner random vectors is given first. This definition involves the commutators of their semi--quantum operators. After that we will focus on the $1$-Meixner random vectors, and derive a system of $d$ partial differential equations satisfied by their Laplace transform. We provide a set of necessary conditions for this system to be integrable. We use these conditions to give a complete characterization of all non--degenerate three--dimensional $1$--Meixner random vectors. It must be mentioned that the three--dimensional case produces the first example in which the components of a $1$--Meixner random vector cannot be reduced, via an injective linear transformation, to three independent classic Meixner random variables.
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 point--wise 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.
We generalize the Beckner's type Poincaré inequality \cite{Beckner} to a large class of probability measures on an abstract Wiener space of the form $μ\starν$, 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. Our dimension-independent results are compared with some very recent findings in the literature. In addition, we prove that in the finite dimensional case the class of densities of convolutions measures satisfies a point-wise covariance inequality.
An important connection between the finite dimensional Gaussian Wick product and Lebesgue convolution product 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.
The paper is divided into two parts. In the first part we lay down the foundation for defining the joint annihilation-preservation-creation decomposition of a finite family of, not necessarily commutative random variables, and show that this decomposition is essentially unique. In the second part we show that any two, not necessarily commutative, random variables $X$ and $Y$, for which the vector space spanned by their annihilation, preservation, and creation operators equipped with the bracket given by the commutator, forms a Lie algebra, are equivalent, up to an invertible linear transformation to two independent Meixner random variables with mixed preservation operators. In particular if $X$ and $Y$ commute, then they are equivalent, up to an invertible linear transformation to two independent classic Meixner random variables. To show this we start with a small technical condition called "non-degeneracy".