Ewa Damek

Harmonic analysis on solvable extensions of H-type groups

To each groupN of Heisenberg type one can associate a generalized Siegel domain, which for specialN is a symmetric space. This domain can be viewed as a solvable extensionS =NA ofN endowed with a natural left-invariant Riemannian metric. We prove that the functions onS that depend only on the distance from the identity form a commutative convolution algebra. This makesS an example of a harmonic manifold, not necessarily symmetric. In order to study this convolution algebra, we introduce the notion of “averaging projector” and of the corresponding spherical functions in a more general context. We finally determine the spherical functions for the groupsS and their Martin boundary.

On multidimensional Mandelbrot cascades

Let Z be a random variable with values in a proper closed convex cone , A a random endomorphism of C and N a random integer. We assume that Z, A, N are independent. Given N independent copies of we define a new random variable . Let T be the corresponding transformation on the set of probability measures on C, i.e. T maps the law of Z to the law of . If the matrix has dominant eigenvalue 1, we study existence and properties of fixed points of T having finite non-zero expectation. Existing one-dimensional results concerning T are extended to higher dimensions. In particular we give conditions under which such fixed points of T have multidimensional regular variation in the sense of extreme value theory and we determine the index of regular variation.

HEAVY TAILED SOLUTIONS OF MULTIVARIATE SMOOTHING TRANSFORMS

Let N>1 be a fixed integer and (C1,…,CN,Q) a random element of M(d×d,R)N×Rd. We consider solutions of multivariate smoothing transforms, i.e. random variables R satisfying R=d∑i=1NCiRi+Q where =d denotes equality in distribution, and R,R1,…,RN are independent identically distributed Rd-valued random variables, and independent of (C1,…,CN,Q). We briefly review conditions for the existence of solutions, and then study their asymptotic behaviour. We show that under natural conditions, these solutions exhibit heavy tails. Our results also cover the case of complex valued weights (C1,…,CN).

Martin boundary for homogeneous riemannian manifolds of negative curvature at the bottom of the spectrum

In this paper we treat noncoercive operators on simply connected homogeneous manifolds of negative curvature.

Precise tail asymptotics of fixed points of the smoothing transform with general weights

We consider solutions of the stochastic equation

Large deviations for solutions to stochastic recurrence equations under Kesten’s condition

R=_d\sum_{i=1}^NA_iR_i+B

Harmonic functions on semidirect extensions of type

, where

H

N>1

nilpotent groups

is a fixed constant,

Large deviation estimates for exceedance times of perpetuity sequences and their dual processes

A_i