Information Transmission under Random Emission Constraints

We model the transmission of a message on the complete graph with n vertices and limited resources. The vertices of the graph represent servers that may broadcast the message at random. Each server has a random emission capital that decreases at each emission. Quantities of interest are the number of servers that receive the information before the capital of all the informed servers is exhausted and the exhaustion time. We establish limit theorems (law of large numbers, central limit theorem and large deviation principle), as n tends to infinity, for the proportion of visited vertices before exhaustion and for the total duration. The analysis relies on a construction of the transmission procedure as a dynamical selection of successful nodes in a Galton-Watson tree with respect to the success epochs of the coupon collector problem

Malliavin Calculus and Skorohod Integration for Quantum Stochastic Processes

A derivation operator and a divergence operator are defined on the algebra of bounded operators on the symmetric Fock space over the complexification of a real Hilbert space \eufrak{h} and it is shown that they satisfy similar properties as the derivation and divergence operator on the Wiener space over \eufrak{h}. The derivation operator is then used to give sufficient conditions for the existence of smooth Wigner densities for pairs of operators satisfying the canonical commutation relations. For \eufrak{h}=L^2(\mathbb{R}_+), the divergence operator is shown to coincide with the Hudson-Parthasarathy quantum stochastic integral for adapted integrable processes and with the non-causal quantum stochastic integrals defined by Lindsay and Belavkin for integrable processes.Comment: 28 pages, amsart styl

On Krawtchouk Transforms

Krawtchouk polynomials appear in a variety of contexts, most notably as orthogonal polynomials and in coding theory via the Krawtchouk transform. We present an operator calculus formulation of the Krawtchouk transform that is suitable for computer implementation. A positivity result for the Krawtchouk transform is shown. Then our approach is compared with the use of the Krawtchouk transform in coding theory where it appears in MacWilliams' and Delsarte's theorems on weight enumerators. We conclude with a construction of Krawtchouk polynomials in an arbitrary finite number of variables, orthogonal with respect to the multinomial distribution.Comment: 13 pages, presented at 10th International Conference on Artificial Intelligence and Symbolic Computation, AISC 2010, Paris, France, 5-6 July 201

Integration with respect to the non-commutative fractional Brownian motion

We study the issue of integration with respect to the non-commutative fractional Brownian motion, that is the analog of the standard fractional Brownian in a non-commutative probability setting.When the Hurst index $H$ of the process is stricly larger than $1/2$, integration can be handled through the so-called Young procedure. The situation where $H=1/2$ corresponds to the specific free case, for which an It{\^o}-type approach is known to be possible.When $H<1/2$, rough-path-type techniques must come into the picture, which, from a theoretical point of view, involves the use of some a-priori-defined L{\'e}vy area process. We show that such an object can indeed be \enquote{canonically} constructed for any $H\in (\frac14,\frac12)$. Finally, when $H\leq 1/4$, we exhibit a similar non-convergence phenomenon as for the non-diagonal entries of the (classical) L{\'e}vy area above the standard fractional Brownian

Skorohod and rough integration with respect to the non-commutative fractional Brownian motion

We pursue our investigations, initiated in [8], about stochastic integration with respect to the non-commutative fractional Brownian motion (NC-fBm). Our main objective in this paper is to compare the pathwise constructions of [8] with a Skorohod-type interpretation of the integral. As a first step, we provide details on the basic tools and properties associated with non-commutative Malliavin calculus, by mimicking the presentation of Nualart's celebrated treatise [14]. Then we check that, just as in the classical (commutative) situation, Skorohod integration can indeed be considered in the presence of the NC-fBm, at least for a Hurst index H > 1 4.This finally puts us in a position to state and prove the desired comparison result, which can be regarded as an Itô-Stratonovich correction formula for the NC-fBm

Dynamics random walks on Heisenberg groups

We prove a Guivarc'h law of large numbers and a central limit theorem for dynamic random walks on Heisenberg groups. The limiting distribution is explicitely given. To our knowledge this is the first study of dynamic random walks on non-commutative Lie groups

Phase Retrieval for Probability Distributions on Quantum Groups and Braided Groups

For nilpotent quantum groups [as introduced by Franz et al. (7)], we show that (in sharp contrast to the classical case) the symmetrization $$\mu * \bar \mu$$ of a probability distribution μ and the first moments of μ together determine uniquely the original distribution