Radial Dunkl processes: Existence,uniqueness and hitting time

We give shorter proofs of the following known results: the radial Dunkl process associated with a reduced system and a strictly positive multiplicity function is the unique strong solution for all times t of a stochastic differential equation with a singular drift, the first hitting time of the Weyl chamber by a radial Dunkl process is finite almost surely for small values of the multiplicity function. The proof of the first result allows one to give a positive answer to a conjecture announced by Gallardo–Yor while that of the second shows that the process hits almost surely the wall corresponding to the simple root with a small multiplicity value. To cite this article: N. Demni, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Quaternionic Brownian Windings

Journal of Theoretical Probability - We define and study the three-dimensional windings along Brownian paths in the quaternionic Euclidean, projective and hyperbolic spaces. In particular, the...     

Moments of the Hermitian Matrix Jacobi Process

In this paper, we compute the expectation of traces of powers of the Hermitian matrix Jacobi process for a large enough but fixed size. To proceed, we first derive the semi-group density of its eigenvalues process as a bilinear series of symmetric Jacobi polynomials. Next, we use the expansion of power sums in the Schur polynomial basis and the integral Cauchy–Binet formula in order to determine the partitions having nonzero contributions after integration. It turns out that these are hooks of bounded weight and the sought expectation results from the integral of a product of two Schur functions with respect to a generalized beta distribution. For special values of the parameters on which the matrix Jacobi process depends, the last integral reduces to the Cauchy determinant and we close the paper with the investigation of the asymptotic behavior of the resulting formula as the matrix size tends to infinity.     

Free Jacobi Process Associated with One Projection: Local Inverse of the Flow

We pursue the study started in Demni and Hmidi (Colloq Math 137(2):271–296, 2014) of the dynamics of the spectral distribution of the free Jacobi process associated with one orthogonal projection. More precisely, we use Lagrange inversion formula in order to compute the Taylor coefficients of the local inverse around \(z=0\) of the flow determined in Demni and Hmidi (Colloq Math 137(2):271–296, 2014). When the rank of the projection equals 1/2, the obtained sequence reduces to the moment sequence of the free unitary Brownian motion. For general ranks in (0, 1), we derive a contour integral representation for the first derivative of the Taylor series.     

The Hermitian Jacobi Process: A Simplified Formula for the Moments and Application to Optical Fiber MIMO Channels

Functional Analysis and Its Applications - 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...     

Markov Semi-groups Associated with the Complex Unimodular Group $$Sl(2,{\mathbb {C}})$$ S l ( 2 , C )

In this paper, we derive the explicit expressions of the Markov semi-groups constructed by Biane (ESAIM Probab Stat 15:S2–S10, 2011) from the restriction of a particular positive definite function on the complex unimodular group \(SL(2,{\mathbb {C}})\) to two commutative subalgebras of its universal \(C^{\star }\)-algebra. Our computations use Euclidean Fourier analysis together with the generating function of Laguerre polynomials with index \(-\,1\), and yield absolutely-convergent double series representations of the semi-group densities. We also supply some arguments supporting the coincidence, noticed by Biane as well, occurring between the heat kernel on the Heisenberg group and the semi-group corresponding to the intersection of the principal and the complementary series. To this end, we appeal to the metaplectic representation \(Mp(4,{\mathbb {R}})\) and to the Landau operator in the complex plane.

     

Large deviations for statistics of the Jacobi process

This paper aims to derive large deviations for statistics of the Jacobi process already conjectured by M. Zani in her thesis. To proceed, we write in a simpler way the Jacobi semi-group density. Being given by a bilinear sum involving Jacobi polynomials, it differs from Hermite and Laguerre cases by the quadratic form of its eigenvalues. Our attempt relies on subordinating the process using a suitable random time change. This gives a Mehler-type formula whence we recover the desired semi-group density. Once we do, an adaptation of Zani’s result [M. Zani, Large deviations for squared radial Ornstein–Uhlenbeck processes, Stochastic. Process. Appl. 102 (1) (2002) 25–42] to the non-steep case will provide the required large deviations principle.     

Integral representation of the sub-elliptic heat kernel on the complex anti-de Sitter fibration

We derive an integral representation for the subelliptic heat kernel of the complex anti-de Sitter fibration. Our proof is different from the one used in Wang (Potential Anal 45:635–653, 2016) since it appeals to the commutativity of the D’Alembertian and of the Laplacian acting on the vertical variable rather than the analytic continuation of the heat semigroup of the real hyperbolic space. Our approach also sheds the light on the connection between the sub-Laplacian of the above fibration and the so-called generalized Maass Laplacian, and on the role played by the odd dimensional real hyperbolic space.     

Free Jacobi Process

In this paper, we define and study two parameters dependent free processes (λ,θ) called free Jacobi, obtained as the limit of its matrix counterpart when the size of the matrix goes to infinity. The main result we derive is a free SDE analogous to that satisfied in the matrix setting, derived under injectivity assumptions. Once we did, we examine a particular case for which the spectral measure is explicit and does not depend on time (stationary). This allows us to determine easily the parameters range ensuring our injectivity requirements so that our result applies. Then, we show that under an additional condition of invertibility at time t=0, this range extends to the general setting. To proceed, we set a recurrence formula for the moments of the process via free stochastic calculus.     

Generalized Bessel functions of dihedral-type: expression as a series of confluent Horn functions and Laplace-type integral representation

The Ramanujan Journal - In the first part of this paper, we express the generalized Bessel function associated with dihedral systems and a constant multiplicity function as an infinite series of...