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qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expec ..."
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Cited by 73 (3 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
A qdeformation of the Gauss distribution
 J. Math. Phys
, 2000
"... The qdeformed commutation relation aa #  qa # a = 11 for the harmonic oscillator is considered with q # [1, 1]. An explicit representation generalizing the Bargmann representation of analytic functions on the complex plane is constructed. In this representation the distribution of a + a # ..."
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The qdeformed commutation relation aa #  qa # a = 11 for the harmonic oscillator is considered with q # [1, 1]. An explicit representation generalizing the Bargmann representation of analytic functions on the complex plane is constructed. In this representation the distribution of a + a # in the vacuum state is explicitly calculated. This distribution is to be regarded as the natural q deformation of the Gaussian. 1995 PACS numbers: 02.50.Cw, 05.40.+j, 03.65.Db, 42.50.Lc 1991 MSC numbers: 81S25, 33D90, 81Q10 1 1
A Mixed QuantumClassical Central Limit Theorem
"... . A randomized qcentral or qcommutative limit theorem on a family of bialgebras with one complex parameter is shown. 1. Introduction In [Spe92] Speicher proved a noncommutative limit theorem, in which the commutation relations of the quantum random variables are given by classical f\Gamma1; 1g ..."
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. A randomized qcentral or qcommutative limit theorem on a family of bialgebras with one complex parameter is shown. 1. Introduction In [Spe92] Speicher proved a noncommutative limit theorem, in which the commutation relations of the quantum random variables are given by classical f\Gamma1; 1gvalued random variables, i.e. Bernoulli random variables. We will show a similar limit theorem, but placed in a bialgebra setting. Here it is the coproduct that depends on a sequence of i.i.d. complexvalued random variables. This also leads to qcommutation relations for the increments, but with less independence in the choice of the commutation factors, as we will see. We will also see that the limit distribution can be expressed as an exponential with respect to a (nonassociative!) averaged convolution. We begin by briefly recalling the results of Speicher[Spe92] and Sch¨urmann[Sch93]. In Secion 2 we introduce the bialgebras used to formulate the main theorem (Section 3), and the algebr...
c World Scientic Publishing Company NONCOMMUTATIVE BROWNIAN MOTIONS ASSOCIATED WITH KESTEN DISTRIBUTIONS AND RELATED POISSON PROCESSES
, 2007
"... We introduce and study a noncommutative twoparameter family of noncommutative Brownian motions in the free Fock space. They are associated with Kesten laws and give a continuous interpolation between Brownian motions in free probability and monotone probability. The combinatorics of our model is ba ..."
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We introduce and study a noncommutative twoparameter family of noncommutative Brownian motions in the free Fock space. They are associated with Kesten laws and give a continuous interpolation between Brownian motions in free probability and monotone probability. The combinatorics of our model is based on ordered noncrossing partitions, in which to each such partition P we assign the weight w(P) = pe(P)qe 0(P), where e(P) and e0(P) are, respectively, the numbers of disorders and orders in P related to the natural partial order on the set of blocks of P implemented by the relation of being inner or outer. In particular, we obtain a simple relation between Delaney's numbers (related to inner blocks in noncrossing partitions) and generalized Euler's numbers (related to orders and disorders in ordered noncrossing partitions). An important feature of our interpolation is that the mixed moments of the corresponding creation and annihilation processes also reproduce their monotone and free counterparts, which does not take place in other interpolations. The same combinatorics is used to construct an interpolation between free and monotone Poisson processes.