The Five Independences as Natural Products

Muraki, Naofumi

doi:10.1142/s0219025703001365

Cited by 99 publications

(87 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The Boolean theory is much simpler than free probability theory, and at this point lacks its depth, primarily because of the lack of the random matrix techniques. 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%

Appell polynomials and their relatives II. Boolean theory

Anshelevich¹

2009

Indiana Univ. Math. J.

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

Appell polynomials and their relatives II. Boolean theory

Anshelevich¹

2009

Indiana Univ. Math. J.

View full text Add to dashboard Cite

show abstract

“…They are denoted by (T) for tensor, (F) for free, and (B) for Boolean corresponding to the tensor, free and Boolean independence respectively. Recently, N. Muraki [13,14] proved that by dropping the commutativity axiom there exist five products including (T), (F) and (B), and the new monotone (M) and anti-monotone (AM) products [10,12]. In section 2 we will consider positive universal products, i.e.…”

Section: Introductionmentioning

confidence: 99%

Quantum Lévy processes on dual groups

Ghorbal

Schürmann

2005

Math. Z.

View full text Add to dashboard Cite

We present a theory of quantum (non-commutative) Lévy processes on dual groups which generalizes the theory of Lévy processes on bialgebras. It follows from a result of N. Muraki that there exist exactly 5 notions of non-commutative 'positive' stochastic independence. We show that one can associate a commutative bialgebra with each pair consisting of a dual group and one of the 5 notions of independence. This construction is related to a construction of U. Franz. Our construction has the advantage that the important case of free independence is included. We show that Lévy processes are given by their generators which are precisely the conditonally positive linear functionals on the dual group.

show abstract

“…Muraki 20,19 has proven that there exist only five universal notions of independence in quantum probability. These are tensor, free, boolean, monotone, and anti-monotone independence.…”

Section: Introductionmentioning

confidence: 99%

Markov Property of Monotone Lévy Processes

Franz¹,

Muraki²

2005

Infinite Dimensional Harmonic Analysis III

Self Cite

View full text Add to dashboard Cite

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.

show abstract

The Five Independences as Natural Products

Cited by 99 publications

References 17 publications

Appell polynomials and their relatives II. Boolean theory

Appell polynomials and their relatives II. Boolean theory

Quantum Lévy processes on dual groups

Markov Property of Monotone Lévy Processes

Contact Info

Product

Resources

About