Yves Guivarc’h

Densité d’orbites d’actions de groupes linéaires et propriétés d’équidistribution de marches aléatoires

Nous considerons un sous-groupe Γ du groupe lineaire G = Sl(d, ℝ), le sous-groupe N des matrices unipotentes triangulaires superieures et le sous-groupe A des matrices diagonales positives. Le sous-groupe Γ est suppose discret et ≪ non elementaire ≫. En utilisant plusieurs notions de points limites pour Γ, nous etudions la densite des orbites de Γ dans certains fermes Γ-invariants canoniques de ℝ d , et de ses produits exterieurs. Nous considerons aussi, par dualite, l’action de N sur Γ\G, puis l’action de A sur Γ\G. Nous utilisons une methode basee sur les proprietes d’equidistribution de marches aleatoires sur ℝ d ou G/N.

Stable laws and spectral gap properties for affine random walks

We consider a general multidimensional affine recursion with corresponding Markov operator P and a unique P-stationary measure. We show spectral gap properties on Holder spaces for the corresponding Fourier operators and we deduce convergence to stable laws for the Birkhoff sums along the recursion. The parameters of the stable laws are expressed in terms of basic quantities depending essentially on the matricial multiplicative part of P. Spectral gap properties of P and homogeneity at infinity of the P-stationary measure play an important role in the proofs.

Homogeneity at Infinity of Stationary Solutions of Multivariate Affine Stochastic Recursions

We consider a d-dimensional affine stochastic recursion of general type corresponding to the relation

Polynomial Growth, Recurrence and Ergodicity for Random Walks on Locally Compact Groups and Homogeneous Spaces

\displaystyle{ X_{n+1} = A_{n+1}X_{n} + B_{n+1},\quad X_{0} = x. }

Harnack Inequality, Martin’s Method and The Positive Spectrum for Random Walks

(S) Under natural conditions, this recursion has a unique stationary solution R, which is unbounded. If d > 2, we sketch a proof of the fact that R belongs to the domain of attraction of a stable law which depends essentially of the linear part of the recursion. The proof is based on renewal theorems for products of random matrices, radial Fourier analysis in the vector space \({\mathbb{R}}^{d}\), and spectral gap properties for convolution operators on the corresponding projective space. We state the corresponding simpler result for d = 1.

Random Walks and Ground State Properties

We describe certain sufficient conditions for an infinitely divisible probability measure on a Lie group to be embeddable in a continuous one-parameter semigroup of probability measures. A major class of Lie groups involved in the analysis consists of central extensions of almost algebraic groups by compactly generated abelian groups without vector part. This enables us in particular to conclude the embeddability of all infinitely divisible probability measures on certain connected Lie groups, including the so called Walnut group. The embeddability is concluded also under certain other conditions. Our methods are based on a detailed study of actions of certain nilpotent groups on special spaces of probability measures and on Fourier analysis along the fibering of the extension.

Subalgebras and Parabolic Subgroups

Let G be a locally compact group, E a homogeneous space of G. We discuss the relations between recurrence of a random walk on G or E, ergodicity of the corresponding transformations and polynomial growth of G or E. We consider the special case of linear groups over local fields.

The Satake-Furstenberg Compactifications

This compactification was introduced by Karpelevic in [K3]. The original inductive definition of \({\overline X ^K}\) is recalled in § 5.3. By examining the closure of a flat A · o in \({\overline X ^K}\), a non-inductive characterization of the closure \({\overline {{A^ + } \cdot o} ^K}\) of \({\overline {A \cdot o} ^K}\) is obtained (see Theorem 5.6). The nature of the Karpelevic topology restricted to the flat is clarified by the introduction of the class of K- fundament al sequences. Using this concept, one shows that (mathtype) is isomorphic to a compactification of a determined by its polyhedral structure. This compactification of a, referred to as the Karpelevic compactification of a, is used to give a new proof that the Karpelevic topology is compact. Lemma 5.26, Proposition 5.27, and Corollary 5.28 explain the relations between the Karpelevic compactification and the conic and dual cell compactifications. Finally, in Remark 5.32 a new way to define the Karpelevic compactification is presented. It consists of fitting together the Karpelevic compactifications of the flats kA · o, k ⊀ K, in exactly the same way that the dual cell compactification is obtained from the polyhedral compactifications of the flats kA · o.