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30
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 98 (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].
Stochastic Calculus With Respect To Free Brownian Motion And Analysis On Wigner Space
, 1998
"... . We define stochastic integrals with respect to free Brownian motion, and show that they satisfy BurkholderGundy type inequalities in operator norm. We prove also a version of Ito's predictable representation theorem, as well as product form and functional form of Ito's formula. Finally ..."
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Cited by 54 (3 self)
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. We define stochastic integrals with respect to free Brownian motion, and show that they satisfy BurkholderGundy type inequalities in operator norm. We prove also a version of Ito's predictable representation theorem, as well as product form and functional form of Ito's formula. Finally we develop stochastic analysis on the free Fock space, in analogy with stochastic analysis on the Wiener space. Introduction In this paper we develop a stochastic integration theory with respect to the free Brownian motion. A strong motivation for undertaking this work was provided by two sources. On one hand the stochastic quantization approach to Master Fields, as described in [D], requires the development of a stochastic calculus with respect to free Brownian motion, in order to be implemented in a mathematically rigourous way. On the other hand, the theory of free entropy developped by D. Voiculescu suggests the study of "free" Gibbs states, whose definition is analogous to the classical Gibbs st...
Large deviation upper bounds and central limit theorems for band matrices and noncommutative functionnals of Gaussian large random matrices
, 2002
"... ABSTRACT. – We obtain large deviation upper bounds and central limit theorems for noncommutative functionals of large Gaussian band matrices and deterministic diagonal matrices with converging spectral measure. As a consequence, we derive such type of results for the spectral measure of Gaussian ba ..."
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Cited by 29 (7 self)
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ABSTRACT. – We obtain large deviation upper bounds and central limit theorems for noncommutative functionals of large Gaussian band matrices and deterministic diagonal matrices with converging spectral measure. As a consequence, we derive such type of results for the spectral measure of Gaussian band matrices and Gaussian sample covariance matrices. 2002 Éditions scientifiques et médicales Elsevier SAS AMS classification: 60F10; 15A52; 60F05
Free Diffusions, Free Entropy And Free Fisher Information
"... . Motivated by the stochastic quantization approach to large N matrix models, we study solutions to free stochastic differential equations dX t = dS t \Gamma 1 2 f(X t )dt where S t is a free brownian motion. We show existence, uniqueness and Markov property of solutions. We define a relative free ..."
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Cited by 26 (0 self)
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. Motivated by the stochastic quantization approach to large N matrix models, we study solutions to free stochastic differential equations dX t = dS t \Gamma 1 2 f(X t )dt where S t is a free brownian motion. We show existence, uniqueness and Markov property of solutions. We define a relative free entropy as well as a relative free Fisher information, and show that these quantities behave as in the classical case. Finally we show that, in contrast with classical diffusions, in general the asymptotic distribution of the free diffusion does not converge, as t ! 1, towards the master field (i.e. the Gibbs state). 1. Introduction The purpose of this paper is to start the study of diffusion equations where the driving noise is a free brownian motion. Reasons for considering such equations will be explained in the next sections of this introduction. 1.1 Gibbs states and diffusion theory. Let V be a C 2 function on R d , with Z = Z R d e \GammaV (x) dx ! 1: The probability measur...
Large Deviations Upper Bounds and Non Commutative Entropies for Some Matrices Ensembles
, 2000
"... ..."
Free stochastic measures via noncrossing partitions
 Adv. Math
"... We show that for stochastic processes with freely independent increments, the partitiondependent stochastic measures canbe expressed purely interms of the higher stochastic measures and the higher diagonal measures of the original process. 1. Introduction. Starting with an operatorvalued stochasti ..."
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Cited by 13 (3 self)
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We show that for stochastic processes with freely independent increments, the partitiondependent stochastic measures canbe expressed purely interms of the higher stochastic measures and the higher diagonal measures of the original process. 1. Introduction. Starting with an operatorvalued stochastic process with freely independent increments X(t), in [A] we defined two families {Prπ} and {Stπ} indexed by set partitions.These objects give a precise meaning to the following heuristic expressions.For a partition π =(B1,B2,...,Bn) ∈P(k), temporarily
Free Jacobi processes
, 2006
"... Abstract. In this paper, we define and study free Jacobi processes of parameters λ> 0 and 0 < θ ≤ 1, as the limit of the complex version of the matrix Jacobi process already defined by Y. Doumerc. In the first part, we focus on the stationary case for which we compute the law (that does not de ..."
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Cited by 8 (2 self)
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Abstract. In this paper, we define and study free Jacobi processes of parameters λ> 0 and 0 < θ ≤ 1, as the limit of the complex version of the matrix Jacobi process already defined by Y. Doumerc. In the first part, we focus on the stationary case for which we compute the law (that does not depend on time) and derive, for λ ∈]0, 1] and 1/θ ≥ λ + 1 a free SDE analogous to the classical one. In the second part, we generalize this result under an additional condition. To proceed, we set a recurrence formula for the moments of the process using free stochastic calculus. This will also be used to compute the p. d. e. satisfied by the Cauchy transform of the free Jacobi’s law. 1.
Microstates free entropy and cost of equivalence relations
 Duke Math. J
"... We define an analog of Voiculescu’s free entropy for ntuples of unitaries u1,..., un in a tracial von Neumann algebra M which normalize a unital subalgebra L ∞ [0, 1] = B ⊂ M. Using this quantity, we define the free dimension δ0(u1,..., un ≬ B). This number depends on u1,..., un only up to orbit eq ..."
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Cited by 8 (2 self)
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We define an analog of Voiculescu’s free entropy for ntuples of unitaries u1,..., un in a tracial von Neumann algebra M which normalize a unital subalgebra L ∞ [0, 1] = B ⊂ M. Using this quantity, we define the free dimension δ0(u1,..., un ≬ B). This number depends on u1,..., un only up to orbit equivalence over B. In particular, if R is a measurable equivalence relation on [0, 1] generated by n automorphisms α1,..., αn, let u1,..., un be the unitaries implementing α1,..., αn in the FeldmanMoore crossed product algebra M = W ∗ ([0, 1], R) ⊃ B = L ∞ [0, 1]. Then the number δ(R) = δ0(u1,..., un ≬ B) is an invariant of the equivalence relation R. If R is treeable, δ(R) coincides with the cost C(R) of R in the sense of D. Gaboriau. In particular, it is n for an equivalence relation induced by a free action of the free group Fn. For a general equivalence relation R possessing a finite graphing of finite cost, δ(R) ≤ C(R). Using the notion of free dimension, we define a dynamical entropy invariant for an automorphism of a measurable equivalence relation (or, more generally, of an rdiscrete measure groupoid) and give examples. 1.
Orbital approach to microstate free entropy, preprint
, 2007
"... Abstract. Motivated by Voiculescu’s liberation theory, we introduce the orbital free entropy χorb for noncommutative selfadjoint random variables (also for “hyperfinite random multivariables”). Besides its basic properties the relation of χorb with the usual free entropy χ is shown. Moreover, the ..."
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Cited by 7 (4 self)
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Abstract. Motivated by Voiculescu’s liberation theory, we introduce the orbital free entropy χorb for noncommutative selfadjoint random variables (also for “hyperfinite random multivariables”). Besides its basic properties the relation of χorb with the usual free entropy χ is shown. Moreover, the dimension counterpart of χorb is discussed.
CENTRAL LIMIT THEOREM FOR THE HEAT KERNEL MEASURE ON THE UNITARY GROUP
, 2009
"... We prove that for a finite collection of realvalued functions f1,..., fn on the group of complex numbers of modulus 1 which are derivable with Lipschitz continuous derivative, the distribution of (trf1,..., trfn) under the properly scaled heat kernel measure at a given time on the unitary group U( ..."
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Cited by 6 (0 self)
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We prove that for a finite collection of realvalued functions f1,..., fn on the group of complex numbers of modulus 1 which are derivable with Lipschitz continuous derivative, the distribution of (trf1,..., trfn) under the properly scaled heat kernel measure at a given time on the unitary group U(N) has Gaussian fluctuations as N tends to infinity, with a covariance for which we give a formula and which is of order N −1. In the limit where the time tends to infinity, we prove that this covariance converges to that obtained by P. Diaconis and S. Evans in a previous work on uniformly distributed unitary matrices. Finally, we discuss some combinatorial aspects of our results.