On a functional equation appearing in characterization of distributions by the optimality of an estimate

Let $X$ be a second countable locally compact Abelian group containing no subgroup topologically isomorphic to the circle group $\mathbb{T}$. Let $\mu$ be a probability distribution on $X$ such that its characteristic function $\hat\mu(y)$ does not vanish and $\hat\mu(y)$ for some $n \geq 3$ satisfies the equation $$\prod_{j=1}^{n} \hat\mu(y_j + y) = \prod_{j=1}^{n}\hat\mu(y_j - y), \quad \sum_{j=1}^{n} y_j = 0, \quad y_1,\dots,y_n,y \in Y.$$ Then $\mu$ is a convolution of a Gaussian distribution and a distribution supported in the subgroup of $X$ generated by elements of order 2

A Phase Transition for Circle Maps and Cherry Flows

We study $C^{2}$ weakly order preserving circle maps with a flat interval. The main result of the paper is about a sharp transition from degenerate geometry to bounded geometry depending on the degree of the singularities at the boundary of the flat interval. We prove that the non-wandering set has zero Hausdorff dimension in the case of degenerate geometry and it has Hausdorff dimension strictly greater than zero in the case of bounded geometry. Our results about circle maps allow to establish a sharp phase transition in the dynamics of Cherry flows

Convolution of Orbital Measures on Symmetric Spaces of type Cp and Dp

We study the absolute continuity of the convolution δ♮eX⋆δ♮eY of two orbital measures on the symmetric spaces SO0(p,p)/SO(p)×SO(p), SU(p,p)/S(U(p)×U(p)) and Sp(p,p)/Sp(p)×Sp(p). We prove sharp conditions on X, Y∈a for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions

Moments of a single entry of circular orthogonal ensembles and Weingarten calculus

Consider a symmetric unitary random matrix $V=(v_{ij})_{1 \le i,j \le N}$ from a circular orthogonal ensemble. In this paper, we study moments of a single entry $v_{ij}$. For a diagonal entry $v_{ii}$ we give the explicit values of the moments, and for an off-diagonal entry $v_{ij}$ we give leading and subleading terms in the asymptotic expansion with respect to a large matrix size $N$. Our technique is to apply the Weingarten calculus for a Haar-distributed unitary matrix.Comment: 17 page

Fine structure of the complex hyperbolic Brownian motion and Rudin’s question

We investigate the fine structure of the complex hyperbolicBrownian motion in the unit ball of Cn. It turns out that the generator of the process is locally very close to the generator of some simple transformation of the classical Brownian motion. This fact helps us to give an intuitive explanation why the invariant Laplace operator in the unit ball of Cn is a difference of two ordinary Laplace operators – the question set by W. Rudin in his monograph Function Theory in the Unit Ball of Cn. In the second part of the paper we find stochastic differential equations for the complex hyperbolic Brownian motion on the ball model of the complex hyperbolic space and furnish in this way an important tool in a further investigation of this process

Familles orthogonales et semigroupes en analyse et probabilités

The Workshop CIMPA-UNESCO ``Orthogonal families and semigroups in analysis and probability\u27\u27 was held in 2006 in Mérida, Venezuela and was organized with the collaboration of three Venezuelian universities(UCV, USB and ULA). The objective of the Workshop was to present the modern theory of operator semigroups, related to polynomial orthogonal expansions. This theory comprises nowadays a vast body of knowledge and has interconnections with several other areas, including harmonic analysis, probability, random matrices, stochastic calculus and control theory. The chapters of this volume originate from the lectures of this Workshop and they stress the interplay of all these domains