Monotone and boolean unitary Brownian motions

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 booleen 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.Comment: 19 page

Stochastic analysis for obtuse random walks

We present a construction of the basic operators of stochastic analysis (gradient and divergence) for a class of discrete-time normal martingales called obtuse random walks. The approach is based on the chaos representation property and discrete multiple stochastic integrals. We show that these operators satisfy similar identities as in the case of the Bernoulli randoms walks. We prove a Clark-Ocone-type predictable representation formula, obtain two covariance identities and derive a deviation inequality. We close the exposition by an application to option hedging in discrete time.Comment: 26 page

Summing free unitary Brownian motions with applications to quantum information

Motivated by quantum information theory, we introduce a dynamical random state built out of the sum of $k \geq 2$ independent unitary Brownian motions. In the large size limit, its spectral distribution equals, up to a normalising factor, that of the free Jacobi process associated with a single self-adjoint projection with trace $1/k$. Using free stochastic calculus, we extend this equality to the radial part of the free average of $k$ free unitary Brownian motions and to the free Jacobi process associated with two self-adjoint projections with trace $1/k$, provided the initial distributions coincide. In the single projection case, we derive a binomial-type expansion of the moments of the free Jacobi process which extends to any $k \geq 3$ the one derived in \cite {DHH} in the special case $k=2$. Doing so give rise to a non normal (except for $k=2$) operator arising from the splitting of a self-adjoint projection into the convex sum of $k$ unitary operators. This binomial expansion is then used to derive a pde for the moment generating function of this non normal operator and for which we determine the corresponding characteristic curves.Comment: The characteristic curves are determine

The Hermitian Jacobi process: simplified formula for the moments and application to optical fibers MIMO channels

Using a change of basis in the algebra of symmetric functions, we compute the moments of the Hermitian Jacobi process. After a careful arrangement of the terms and the evaluation of the determinant of an `almost upper-triangular' matrix, we end up with a moment formula which is considerably simpler than the one derived in \cite{Del-Dem}. As an application, we propose the Hermitian Jacobi process as a dynamical model for optical fibers MIMO channels and compute its Shannon capacity for small enough power at the transmitter. Moreover, when the size of the Hermitian Jacobi process is larger than the moment order, our moment formula may be written as a linear combination of balanced terminating ${}_4F_3$-series evaluated at unit argument.Comment: Accepted for publication in Funct. Anal. Appl

The quantum mechanics canonically associated to free probability Part I: Free momentum and associated kinetic energy

After a short review of the quantum mechanics canonically associated with a classical real valued random variable with all moments, we begin to study the quantum mechanics canonically associated to the \textbf{standard semi--circle random variable} $X$, characterized by the fact that its probability distribution is the semi--circle law $\mu$ on $[-2,2]$. We prove that, in the identification of $L^2([-2,2],\mu)$ with the $1$--mode interacting Fock space $\Gamma_{\mu}$, defined by the orthogonal polynomial gradation of $\mu$, $X$ is mapped into position operator and its canonically associated momentum operator $P$ into $i$ times the $\mu$--Hilbert transform $H_{\mu}$ on $L^2([-2,2],\mu)$. In the first part of the present paper, after briefly describing the simpler case of the $\mu$--harmonic oscillator, we find an explicit expression for the action, on the $\mu$--orthogonal polynomials, of the semi--circle analogue of the translation group $e^{itP}$ and of the semi--circle analogue of the free evolution $e^{itP^2/2}$ respectively in terms of Bessel functions of the first kind and of confluent hyper--geometric series. These results require the solution of the \textit{inverse normal order problem} on the quantum algebra canonically associated to the classical semi--circle random variable and are derived in the second part of the present paper. Since the problem to determine, with purely analytic techniques, the explicit form of the action of $e^{-tH_{\mu}}$ and $e^{-itH_{\mu}^2/2}$ on the $\mu$--orthogonal polynomials is difficult, % aaa ask T if it is solved the above mentioned results show the power of the combination of these techniques with those developed within the algebraic approach to the theory of orthogonal polynomials.Comment: 28 page

Analyse expérimentale, par PIV, de la corrélation entre la dynamique tourbillonnaire d'un jet heurtant une plaque fendue et le champ acoustique généré

International audienceLors de l’impact entre un jet et une plaque fendue, des nuisances sonores peuvent être générées. Réellement,au niveau de la fente, l’obstacle peut interagir avec l’écoulement, sous certaines conditions, pour donner naissance à des perturbations de la dynamique tourbillonnaire de l’écoulement et peut contrôler le détachement tourbillonnaire dès sa naissance. Ces perturbations dynamiques mettent en relief le transfert d’énergie du champ aérodynamique vers le champ acoustique ainsi généré. Afin de visualiser la partie tourbillonnaire de l’écoulement et de mettre en exergue les mécanismes responsables des nuisances sonores, une métrologie laser a été appliquée : la PIV (vélocimétrie par images de particules). Cette méthode de mesures est basée sur l’ensemencement de l’écoulement avec des traceurs appropriés dans l’écoulement, et à la génération de plans lumineux, à l’aide d’un laser pulsé à deux têtes, afin de matérialiser les plans étudiés. Une caméra rapide phantom V711 est utilisée pour enregistrer les paires d’images ainsi réalisées. Un logiciel dédié à laPIV « DaVis » permet de réaliser les calculs de champs cinématiques en fonction du temps. Dans cette étude,on met en évidence les corrélations qui existent entre les pics acoustiques observés dans certaines configurations et la dynamique tourbillonnaire de l’écoulement. Pour cela des microphones Brüel & Kjær et une centrale d’acquisition équipée d’une carte NI PXI – 4472 ont été utilisés pour la partie acoustique, et un suivi spécifique des structures tourbillonnaires a été réalisé