Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Aspects of Convergence of Random Walks on Finite Volume Homogeneous Spaces

Roland Prohaska

Abstract

We investigate three aspects of weak* convergence of the $n$-step distributions of random walks on finite volume homogeneous spaces $G/Γ$ of semisimple real Lie groups. First, we look into the obvious obstruction to the upgrade from Cesaro to non-averaged convergence: periodicity. We give examples where it occurs and conditions under which it does not. In a second part, we prove convergence towards Haar measure with exponential speed from almost every starting point. Finally, we establish a strong uniformity property for the Cesaro convergence towards Haar measure for uniquely ergodic random walks.

Aspects of Convergence of Random Walks on Finite Volume Homogeneous Spaces

Abstract

We investigate three aspects of weak* convergence of the -step distributions of random walks on finite volume homogeneous spaces of semisimple real Lie groups. First, we look into the obvious obstruction to the upgrade from Cesaro to non-averaged convergence: periodicity. We give examples where it occurs and conditions under which it does not. In a second part, we prove convergence towards Haar measure with exponential speed from almost every starting point. Finally, we establish a strong uniformity property for the Cesaro convergence towards Haar measure for uniquely ergodic random walks.

Paper Structure

This paper contains 12 sections, 16 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Periodicity
  3. Examples
  4. An Aperiodicity Criterion
  5. Spectral Gap
  6. Generic Points
  7. Good Height Functions
  8. Sobolev Norms
  9. Exponentially Generic Points
  10. Uniform Cesàro Convergence
  11. Locally Uniform Convergence
  12. Lyapunov Functions & Stronger Uniformity

Key Result

Theorem \oldthetheorem

Let $\mu$ be a compactly supported probability measure on $G$. Denote by $\mathcal{S}$ and $\mathcal{G}$ the closed subsemigroup and subgroup of $G$ generated by $\mathop{\mathrm{supp}}\nolimits(\mu)$, respectively, and suppose that the Zariski closure of $\mathop{\mathrm{Ad}}\nolimits(\mathcal{G})$ as $n\to\infty$ in the weak* topology.

Theorems & Definitions (49)

  • Theorem \oldthetheorem: Benoist--Quint BQ3
  • Definition \oldthetheorem
  • Example \oldthetheorem
  • Example \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • proof : Proof of Theorem \ref{['thm:aperiodicity']}
  • ...and 39 more