Jean-François Quint

Stationary measures and invariant subsets of homogeneous spaces (II)

Let G be a real Lie group, be a lattice in G and be a compactly generated closed subgroup of G. If the Zariski closure of the group Ad() is semisimple with no compact factor, we prove that every -orbit closure in G= is a nite volume homogeneous space. We also establish related equidistribution properties.

Central limit theorem for linear groups

We prove a central limit theorem for random walks with finite variance on linear groups.

Random Walks on Reductive Groups

We apply the previous results to random walks in products of algebraic reductive groups over local fields. We prove the Law of Large Numbers for both the Iwasawa cocycle and the Cartan projection. We prove also the regularity of the Lyapunov vector.

Random walks on projective spaces

Let G be a connected real semisimple Lie group, V be a finite dimensional representation of G, and μ be a probability measure on G whose support spans a Zariski dense subgroup. We prove that the set of ergodic μ-stationary probability measures on the projective space P(V ) is in one-to-one correspondance with the set of compact G-orbits in P(V ). When V is strongly irreducible, we prove the existence of limits for the empirical measures. We prove related results over local fields as the finiteness of the set of ergodic μ-stationary measures on the flag variety of G.

Central limit theorem on hyperbolic groups

We prove a central limit theorem for random walks with finite variance on Gromov hyperbolic groups.

Linear Random Walks

We study random walks on the linear groups. We prove the Law of Large Numbers for the norm of matrices and for the norm of vectors. We also prove the positivity of the first Lyapunov exponent.

Zariski Dense Subsemigroups

We study Zariski dense subgroups in products of algebraic reductive groups over local fields.

Reductive Groups and Their Representations

We recall a few basic facts on reductive groups over local fields, their algebraic representations, their flag varieties, their Cartan projection and their Iwasawa cocycle.

Limit Laws for Cocycles

We prove the Central Limit Theorem, the Law of Iterated Logarithm and the Large Deviation Principle for a cocycle over a contracting action.

The Local Limit Theorem for Cocycles

We prove a Local Limit Theorem with moderate deviations for cocycles over a contracting action.