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117
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. ..."
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Cited by 46 (4 self)
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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.
H∞ FUNCTIONAL CALCULUS AND SQUARE FUNCTIONS ON Noncommutative L^Pspaces
, 2006
"... In this work we investigate semigroups of operators acting on noncommutative L pspaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and H ∞ functional calculus. We discuss several examples of noncommutative diffusion semig ..."
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Cited by 33 (12 self)
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In this work we investigate semigroups of operators acting on noncommutative L pspaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and H ∞ functional calculus. We discuss several examples of noncommutative diffusion semigroups. This includes Schur multipliers, qOrnsteinUhlenbeck semigroups, and the noncommutative Poisson semigroup on free groups.
APPROXIMATION PROPERTIES AND ABSENCE OF CARTAN Subalgebra For Free Arakiwoods Factors
, 2011
"... ..."
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 ..."
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Cited by 20 (1 self)
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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.
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. ..."
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Cited by 19 (2 self)
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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.
Quadratic harnesses, qcommutations, and orthogonal martingale polynomials
 Trans. Amer. Math. Soc
, 2007
"... Abstract. We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a qcommutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical ..."
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Cited by 18 (6 self)
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Abstract. We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a qcommutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical constants. Explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest. 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 ..."
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Cited by 17 (1 self)
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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.
The bi  Poisson process: a quadratic harness
 Annals of Probability
"... This paper is a continuation of our previous research on quadratic harnesses, that is, processes with linear regressions and quadratic conditional variances. Our main result is a construction of a Markov process from given orthogonal and martingale polynomials. The construction uses a twoparameter ..."
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Cited by 16 (3 self)
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This paper is a continuation of our previous research on quadratic harnesses, that is, processes with linear regressions and quadratic conditional variances. Our main result is a construction of a Markov process from given orthogonal and martingale polynomials. The construction uses a twoparameter extension of the AlSalam–Chihara polynomials and a relation between these polynomials for different values of parameters. 1. Introduction. The