Unification of boolean, monotone, anti-monotone, and tensor independence and L�vy processes

Franz, Uwe

doi:10.1007/s00209-002-0469-8

Cited by 28 publications

(37 citation statements)

References 21 publications

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“…The preceding proposition is closely related to the Markov property for the processes with Boolean independent increments (see Section 4.2 of [Fra03]), which will be investigated in a future paper.…”

Section: In Particular Boolean Appell Polynomials Are Boolean Martinmentioning

confidence: 99%

“…Nevertheless, it has a number of crucial properties since it comes from a universal product; in fact, by a theorem of Speicher [Spe97] (see also [BGS02,Mur03]), Boolean, free, and classical theories are exactly the only ones which arise from universal products (respectively, Boolean, free, and tensor) which do not depend on the order of the components. A sample of work on Boolean probability theory includes [SW97,Pri01,Ora02,Fra03,KY04,Mło04,Len05,Sto05,Ber06].…”

Section: Introductionmentioning

confidence: 99%

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Appell polynomials and their relatives II. Boolean theory

Anshelevich¹

2009

Indiana Univ. Math. J.

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ABSTRACT. The Appell-type polynomial family corresponding to the simplest non-commutative derivative operator turns out to be connected with the Boolean probability theory, the simplest of the three universal non-commutative probability theories (the other two being free and tensor/classical probability). The basic properties of the Boolean Appell polynomials are described. In particular, their generating function turns out to have a resolvent-type form, just like the generating function for the free Sheffer polynomials. It follows that the Meixner (that is, Sheffer plus orthogonal) polynomial classes, in the Boolean and free theory, coincide. This is true even in the multivariate case. A number of applications of this fact are described, to the Belinschi-Nica and Bercovici-Pata maps, conditional freeness, and the Laha-Lukacs type characterization.A number of properties which hold for the Meixner class in the free and classical cases turn out to hold in general in the Boolean theory. Examples include the behavior of the Jacobi coefficients under convolution, the relationship between the Jacobi coefficients and cumulants, and an operator model for cumulants. Along the way, we obtain a multivariate version of the Stieltjes continued fraction expansion for the moment generating function of an arbitrary state with monic orthogonal polynomials.

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Section: In Particular Boolean Appell Polynomials Are Boolean Martinmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Appell polynomials and their relatives II. Boolean theory

Anshelevich¹

2009

Indiana Univ. Math. J.

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“…But the additive monotone Brownian motion and the monotone Fock space where first studied by Muraki, see [9]. The monotone stochastic calculus can also be realized on the symmetric Fock space, see [5]. Bercovici studied also the boolean convolution for probability measures on the positive half-line, but it is not always defined.…”

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confidence: 99%

Monotone and boolean unitary Brownian motions

Hamdi

2015

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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Abstract. The additive monotone (resp. boolean) unitary Brownian motion is a non-commutative stochastic process with monotone (resp. boolean) independent and stationary increments which are distributed according to the arcsine law (resp. Bernoulli law) . We introduce the monotone and boolean unitary Brownian motions and we derive a closed formula for their associated moments. This provides a description of their spectral measures. We prove that, in the monotone case, the multiplicative analog of the arcsine distribution is absolutely continuous with respect to the Haar measure on the unit circle, whereas in the boolean case the multiplicative analog of the Bernoulli distribution is discrete. Finally, we use quantum stochastic calculus to provide a realization of these processes as the stochastic exponential of the correspending additive Brownian motions. IntroductionIn non-commutative probability theory, there exist several natural notions of independence. The most famous ones are the usual independence and the free independence. Other interesting examples are monotone and boolean independence. These allow to define new convolutions for probability measures. The monotone convolutions on the unit circle and the positive half-line were introduced by Bercovici in [2], see also [6]. But the additive monotone Brownian motion and the monotone Fock space where first studied by Muraki, see [9]. The monotone stochastic calculus can also be realized on the symmetric Fock space, see [5]. Bercovici studied also the boolean convolution for probability measures on the positive half-line, but it is not always defined. For probability measures on the unit circle, the boolean convolution is well defined. It was introduced by Franz in [7], see also [3]. The boolean stochastic calculus has been studied by Ben Ghorbal and Schürmann, see [1].The aim of this paper is to point out several connections between classical, free Brownian motions and their counterparts in the monotone and boolean cases. We shall consider two kinds of unitary Brownian motions, the monotone and boolean one. Both are non-commutative unitary processes, that is families of noncommutative random variables which are unitary and are characterized as having 2010 Mathematics Subject Classification. Primary 60J65, 46L51, 46L53; Secondary 65C30.

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“…These are tensor, free, boolean, monotone, and anti-monotone independence. Franz 10 subsequently found a construction that reduces boolean, monotone, and anti-monotone independence to tensor independence. As an application the theories of quantum Lévy processes with boolean, monotone, and anti-monotone increments can be reduced to the theory of Lévy processes on involutive bialgebras.…”

Section: Introductionmentioning

confidence: 99%

Markov Property of Monotone Lévy Processes

Franz¹,

Muraki²

2005

Infinite Dimensional Harmonic Analysis III

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Monotone Lévy processes with additive increments are defined and studied. It is shown that these processes have natural Markov structure and their Markov transition semigroups are characterized using the monotone Lévy-Khintchine formula. 17 Monotone Lévy processes turn out to be related to classical Lévy processes via Attal's "remarkable transformation." A monotone analogue of the family of exponential martingales associated to a classical Lévy process is also defined.

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Unification of boolean, monotone, anti-monotone, and tensor independence and L�vy processes

Cited by 28 publications

References 21 publications

Appell polynomials and their relatives II. Boolean theory

Appell polynomials and their relatives II. Boolean theory

Monotone and boolean unitary Brownian motions

Markov Property of Monotone Lévy Processes

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