Results 1  10
of
56
Free martingale polynomials
 Journal of Functional Analysis
"... ABSTRACT. In this paper we investigate the properties of the free Sheffer systems, which are certain families of martingale polynomials with respect to the free Lévy processes. First, we classify such families that consist of orthogonal polynomials; these are the free analogs of the Meixner systems. ..."
Abstract

Cited by 28 (3 self)
 Add to MetaCart
ABSTRACT. In this paper we investigate the properties of the free Sheffer systems, which are certain families of martingale polynomials with respect to the free Lévy processes. First, we classify such families that consist of orthogonal polynomials; these are the free analogs of the Meixner systems. Next, we show that the fluctuations around free convolution semigroups have as principal directions the polynomials whose derivatives are martingale polynomials. Finally, we indicate how Rota’s finite operator calculus can be modified for the free context.
Appell polynomials and their relatives
 Int. Math. Res. Not
, 2004
"... ABSTRACT. This paper summarizes some known results about Appell polynomials and investigates their various analogs. The primary of these are the free Appell polynomials. In the multivariate case, they can be considered as natural analogs of the Appell polynomials among polynomials in noncommuting va ..."
Abstract

Cited by 15 (1 self)
 Add to MetaCart
ABSTRACT. This paper summarizes some known results about Appell polynomials and investigates their various analogs. The primary of these are the free Appell polynomials. In the multivariate case, they can be considered as natural analogs of the Appell polynomials among polynomials in noncommuting variables. They also fit well into the framework of free probability. For the free Appell polynomials, a number of combinatorial and “diagram ” formulas are proven, such as the formulas for their linearization coefficients. An explicit formula for their generating function is obtained. These polynomials are also martingales for free Lévy processes. For more general free Sheffer families, a necessary condition for pseudoorthogonality is given. Another family investigated are the KailathSegall polynomials. These are multivariate polynomials, which share with the Appell polynomials nice combinatorial properties, but are always orthogonal. Their origins lie in the Fock space representations, or in the theory of multiple stochastic integrals. Diagram formulas are proven for these polynomials as well, even in the qdeformed case. 1.
Strong Haagerup inequalities for free Rdiagonal elements
 J. FUNCT. ANAL
, 2007
"... In this paper, we generalize Haagerup’s inequality [H] (on convolution norm in the free group) to a very general context of Rdiagonal elements in a tracial von Neumann algebra; moreover, we show that in this “holomorphic” setting, the inequality is greatly improved from its originial form. We give ..."
Abstract

Cited by 11 (6 self)
 Add to MetaCart
In this paper, we generalize Haagerup’s inequality [H] (on convolution norm in the free group) to a very general context of Rdiagonal elements in a tracial von Neumann algebra; moreover, we show that in this “holomorphic” setting, the inequality is greatly improved from its originial form. We give combinatorial proofs of two important special cases of our main result, and then generalize these techniques. En route, we prove a number of moment and cumulant estimates for Rdiagonal elements that are of independent interest. Finally, we use our strong Haagerup inequality to prove a strong ultracontractivity theorem, generalizing and improving the one in [Bi2].
Popa M.: Feynman Diagrams and Wick products associated with qFock space
 Proc. Natl. Acad. Sci. USA 100
, 2003
"... Abstract. It is shown that if one keeps track of crossings, Feynman diagrams can be used to compute qWick products and normal products in terms of each other. 1. ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
Abstract. It is shown that if one keeps track of crossings, Feynman diagrams can be used to compute qWick products and normal products in terms of each other. 1.
qLévy processes
 J. Reine Angew. Math
, 2004
"... ABSTRACT. We continue the investigation of the Lévy processes on a qdeformed full Fock space started in [1]. First, we show that the vacuum vector is cyclic and separating for the algebra generated by such a process. Next, we describe a chaotic representation property in terms of multiple integrals ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
ABSTRACT. We continue the investigation of the Lévy processes on a qdeformed full Fock space started in [1]. First, we show that the vacuum vector is cyclic and separating for the algebra generated by such a process. Next, we describe a chaotic representation property in terms of multiple integrals with respect to diagonal measures, in the style of Nualart and Schoutens. We define stochastic integration with respect to these processes, and calculate their combinatorial stochastic measures. Finally, we show that they generate infinite von Neumann algebras. 1.
Free cumulants and enumeration of connected partitions
 Eur. J. Combin
"... Abstract. A combinatorial formula is derived which expresses free cumulants in terms of classical comulants. As a corollary, we give a combinatorial interpretation of free cumulants of classical distributions, notably Gaussian and Poisson distributions. The latter count connected pairings and connec ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
Abstract. A combinatorial formula is derived which expresses free cumulants in terms of classical comulants. As a corollary, we give a combinatorial interpretation of free cumulants of classical distributions, notably Gaussian and Poisson distributions. The latter count connected pairings and connected partitions respectively. The proof relies on Möbius inversion on the partition lattice. 1.
Conditional moments of qMeixner processes
, 2004
"... Abstract. We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a threeparameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these proce ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
Abstract. We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a threeparameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the noncommutative generalizations of the Lévy processes. 1.
Factoriality of qgaussian von neumann algebras
 Comm. Math. Phys
"... We prove that the von Neumann algebras generated by n qGaussian elements, are factors for n � 2. 1 ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
We prove that the von Neumann algebras generated by n qGaussian elements, are factors for n � 2. 1
Rosenthal type inequalities for free chaos
, 2005
"... Let A denote the reduced amalgamated free product of a family ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
Let A denote the reduced amalgamated free product of a family