Results 1  10
of
16
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 expectation ..."
Abstract

Cited by 64 (2 self)
 Add to MetaCart
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].
Spectral measure of large random Hankel, Markov and Toeplitz matrices
 Ann. Probab
"... Abstract. We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure. For Hankel and Toeplitz matrices generated by i.i.d. random variables {Xk} of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables {Xi,j}j>i of zero mea ..."
Abstract

Cited by 23 (8 self)
 Add to MetaCart
Abstract. We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure. For Hankel and Toeplitz matrices generated by i.i.d. random variables {Xk} of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables {Xi,j}j>i of zero mean and unit variance, scaling the eigenvalues by √ n we prove the almost sure, weak convergence of the spectral measures to universal, nonrandom, symmetric distributions γH, γM, and γT of unbounded support. The moments of γH and γT are the sum of volumes of solids related to Eulerian numbers, whereas γM has a bounded smooth density given by the free convolution of the semicircle and normal densities. For symmetric Markov matrices generated by i.i.d. random variables {Xi,j}j>i of mean m and finite variance, scaling the eigenvalues by n we prove the almost sure, weak convergence of the spectral measures to the atomic measure at −m. If m = 0, and the fourth moment is finite, we prove that the spectral norm of Mn scaled by √ 2n log n converges almost surely to one. 1. Introduction and
Cumulants in noncommutative probability theory. I. Noncommutative exchangeability systems
 Math. Z
"... Abstract. Cumulants linearize convolution of measures. We use a formula of Good to define noncommutative cumulants in a very general setting. It turns out that the essential property needed is exchangeability of random variables. Roughly speaking the formula says that cumulants are moments of a cert ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
Abstract. Cumulants linearize convolution of measures. We use a formula of Good to define noncommutative cumulants in a very general setting. It turns out that the essential property needed is exchangeability of random variables. Roughly speaking the formula says that cumulants are moments of a certain “discrete Fourier transform ” of a random variable. This provides a simple unified method to understand the known examples of
Symmetric Hilbert spaces arising from species of structures
"... Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces over some `one particle space' K are formed by certain symmetrization procedures performed on the full Fock space. We investigate alternative ways of symmetrization by building on Joyal's notion of a combinatorial species. An ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces over some `one particle space' K are formed by certain symmetrization procedures performed on the full Fock space. We investigate alternative ways of symmetrization by building on Joyal's notion of a combinatorial species. Any such species F gives rise to an endofunctor F of the category of Hilbert spaces with contractions mapping a Hilbert space K to a symmetric Hilbert space F (K) with the same symmetry as the species F . A general framework for annihilation and creation operators on these spaces is developed, and compared to the generalised Brownian motions of R. Speicher and M. Bo _ zejko. As a corollary we nd that the commutation relation a i a j a j a i = f(N) ij with Na i a i N = a i admits a realization on a symmetric Hilbert space whenever f has a power series with infinite radius of convergence and positive coefficients.
Generalised Brownian Motion and Second Quantisation
, 2000
"... A new approach to the generalised Brownian motion introduced by M. ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
A new approach to the generalised Brownian motion introduced by M.
Fourth Moment Theorem and qBrownian Chaos
 Comm. Math. Phys
, 2012
"... Abstract: In 2005, Nualart and Peccati [13] proved the socalled Fourth Moment Theorem asserting that, for a sequence of normalized multiple WienerItô integrals to converge to the standard Gaussian law, it is necessary and sufficient that its fourth moment tends to 3. A few years later, Kemp et al. ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
Abstract: In 2005, Nualart and Peccati [13] proved the socalled Fourth Moment Theorem asserting that, for a sequence of normalized multiple WienerItô integrals to converge to the standard Gaussian law, it is necessary and sufficient that its fourth moment tends to 3. A few years later, Kemp et al. [9] extended this theorem to a sequence of normalized multiple Wigner integrals, in the context of the free Brownian motion. The qBrownian motion, q ∈ (−1, 1], introduced by the physicists Frisch and Bourret [6] in 1970 and mathematically studied by Bo˙zejko and Speicher [2] in 1991, interpolates between the classical Brownian motion (q = 1) and the free Brownian motion (q = 0), and is one of the nicest examples of noncommutative processes. The question we shall solve in this paper is the following: what does the Fourth Moment Theorem become when dealing with a qBrownian motion?
Functors of White Noise Associated to Characters of the Infinite Symmetric Group
, 2001
"... The characters ; of the in nite symmetric group are extended to multiplicative positive de nite functions t; on pair partitions by using an explicit representation due to Versik and Kerov. The von Neumann algebra ; (K) generated by the elds !; (f) with f in an in nite dimensional real Hilbert space ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
The characters ; of the in nite symmetric group are extended to multiplicative positive de nite functions t; on pair partitions by using an explicit representation due to Versik and Kerov. The von Neumann algebra ; (K) generated by the elds !; (f) with f in an in nite dimensional real Hilbert space K is in nite and the vacuum vector is not separating. For a family tN depending on an integer N < f j): The algebras N (` 2 R (Z)) are type I1 factors. Functors of white noise N are constructed and proved to be nonequivalent for dierent values of N . 1
On the structure of noncommutative white noises
, 2004
"... Abstract. We consider the concepts of continuous Bernoulli systems and noncommutative white noises. We address the question of isomorphism of continuous Bernoulli systems and show that for large classes of quantum Lévy processes one can make quite precise statements about the time behaviour of thei ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract. We consider the concepts of continuous Bernoulli systems and noncommutative white noises. We address the question of isomorphism of continuous Bernoulli systems and show that for large classes of quantum Lévy processes one can make quite precise statements about the time behaviour of their moments. 1.
COMPUTING MOMENTS OF FREE ADDITIVE CONVOLUTION OF MEASURES
, 2006
"... This short note explains how to use readytouse components of symbolic software to convert between the free cumulants and the moments of measures without sophisticated programming. This allows quick access to low order moments of free convolutions of measures, which can be used to test whether a gi ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
This short note explains how to use readytouse components of symbolic software to convert between the free cumulants and the moments of measures without sophisticated programming. This allows quick access to low order moments of free convolutions of measures, which can be used to test whether a given probability measure is a free convolution of other measures.
Itô calculus and quantum white noise calculus
"... Summary. Itô calculus has been generalized in white noise analysis and in quantum stochastic calculus. Quantum white noise calculus is a third generalization, unifying the two above mentioned ones and bringing some unexpected insight into some old problems studied in different fields, such as the re ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Summary. Itô calculus has been generalized in white noise analysis and in quantum stochastic calculus. Quantum white noise calculus is a third generalization, unifying the two above mentioned ones and bringing some unexpected insight into some old problems studied in different fields, such as the renormalization problem in physics and the representation theory of Lie algebras. The present paper is an attempt to explain the motivations of these extensions with emphasis on open challenges. The last section includes a result obtained after the Abel Symposium. Namely that, after introducing a new renormalization technique, the RHPWN Lie algebra includes (in fact we will prove elsewhere that this inclusion is an identification) a second quantized version of the extended Virasoro algebra, i.e. the Virasoro–Zamolodchikov ∗–Lie algebra w∞, which has been widely studied in string theory and in conformal field theory. 3 1