Results 1  10
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21
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 ..."
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Cited by 64 (2 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 we develop ..."
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Cited by 35 (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...
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 19 (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 7 (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
On the Algebraic Foundations of NonCommutative Probability Theory
"... As a first part of a rigorous mathematical theory of noncommutative probability we present, starting from a set of canonical axioms, a complete classification of the notions of noncommutative stochastic independence. Our result originates from a first contribution and a conjecture by M. Schurmann ..."
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Cited by 5 (0 self)
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As a first part of a rigorous mathematical theory of noncommutative probability we present, starting from a set of canonical axioms, a complete classification of the notions of noncommutative stochastic independence. Our result originates from a first contribution and a conjecture by M. Schurmann [21; 22] and is based on a fundamental paper by R. Speicher [26]. Our concept is used to initiate a theory of noncommutative L'evy processes which are defined on dual groups in the sense of D. Voiculescu [29]. The paper generalises L'evy processes on Hopf algebras [20] to noncommutative independences other than the `tensor' independence of R. L. Hudson [9, 12].
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 4 (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.
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 depend o ..."
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Cited by 4 (1 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.
CENTRAL LIMIT THEOREMS FOR THE BROWNIAN MOTION ON LARGE UNITARY GROUPS
, 904
"... Abstract. In this paper, we are concerned with the large N limit of linear combinations of entries of Brownian motions on the group of N × N unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one. Various scales of time are concerned, giving rise to vari ..."
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Cited by 2 (0 self)
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Abstract. In this paper, we are concerned with the large N limit of linear combinations of entries of Brownian motions on the group of N × N unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one. Various scales of time are concerned, giving rise to various limit processes, in relation to the geometric construction of the unitary Brownian motion. As an application, we recover certain results about linear combinations of the entries of Haar distributed random unitary matrices.
βJACOBI PROCESSES
, 901
"... Abstract. We define and study a [0,1] mvalued process depending on three positive real parameters p, q, β that specializes for β = 1,2 to the eigenvalues process of the real and complex matrix Jacobi processes on the one hand and that has the distribution of the βJacobi ensemble as stationary dist ..."
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Abstract. We define and study a [0,1] mvalued process depending on three positive real parameters p, q, β that specializes for β = 1,2 to the eigenvalues process of the real and complex matrix Jacobi processes on the one hand and that has the distribution of the βJacobi ensemble as stationary distribution on the other hand. We first prove that this process, called βJacobi process, is the unique strong solution of the stochastic differential equation defining it provided that β> 0, p ∧ q> m − 1 + 1/β. When specialized to β = 1,2, our results actually improve well known results on eigenvalues of matrix Jacobi processes. While proving the strong uniqueness, the generator of the βJacobi process is mapped into the radial part of the DunklCherednik Laplacian associated with the non reduced root system of type BC. The transformed process is then valued in the principal Weyl alcove and this allows to define the Brownian motion in the Weyl alcove corresponding to all multiplicities equal one. Second, we determine, using stochastic calculus and a comparison theorem, the range of β, p, q for which the m components of the βJacobi process first collide, the smallest one reaches 0 and the largest one reaches 1. This is equivalent to the first hitting time of the boundary of the principal Weyl alcove by the transformed process. Finally, we write down its semi group density. 1.