We consider a nondegenerate random walk on a locally compact group with finite first moment. Then, if there are no nonconstant bounded harmonic functions, all the linear drift comes from a real additive character on the group. As a corollary we obtain a generalization of Varopoulos’ theorem that in the case of symmetric random walks, positive linear drift implies the existence of nonconstant bounded harmonic functions. Another consequence is the phenomenon that for some groups (including certain Grigorchuk groups) the drift vanishes for any measure of finite first moment.

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Linear Drift and Poisson Boundary for Random Walks

Ergodic Theory of ℤ d Actions: Boundaries of invariant Markov Operators: The identification problem

We give a lower estimate for the central value μ *n (e) of the nth convolution power μ*...*μ of a symmetric probability measure μ on a polycyclic group G of exponential growth whose suppose is finite and generates G. We also give a similar large time diagonal estimate for the fundamendal solution of the equation (∂/∂t + L)u = 0, where L is a left invariant sub-Laplacian on a unimodular amenable Lie group G of exponential growth

A Lower Estimate for Central Probabilities on Polycyclic Groups

We present recent results about the asymptotic behavior of ergodic products of isometries of a metric space X. If we assume that the displacement is integrable, then either there is a sublinear diffusion or there is, for almost every trajectory in X, a preferred direction at the boundary. We discuss the precise statement when X is a proper metric space and compare it with classical ergodic theorems. Applications are given to ergodic theorems for nonintegrable functions, random walks on groups and Brownian motion on covering manifolds.

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Noncommutative ergodic theorems

Outline of lectures given at the meeting Probabilistic and Dynamical Properties of (semi)-group Actions, held at Universidad de Santiago de Chile, January 2008, and at Wroc÷aw University as part of the EU Marie Curie Host Fellowship program Transfer of Knowledge, June 2008. The author is grateful to Andres Navas and to Ewa Damek for their kind invitations to Santiago and to Wroc÷aw, respectively. Contents 1. Lecture 1: Introduction and Outline 1 2. Lectures 2-3: Horofunctions 5 3. Lecture 4: Proof of the noncommutative ergodic theorem 13 4. Lecture 5: Consequences and ray approximation 17 5. Lecture 6: An application to random walks on groups 21 6. Lecture 7: The Liouville property and drift homomorphisms 24 7. Lecture 8: Extensions of von Neumanns ergodic theorem 27 References 28 1. Lecture 1: Introduction and Outline 1.1. The classical law of large numbers asserts that for i.i.d. random variables Xi with E[jXij] <1, 1 n (X1 +X2 + :::+Xn)! E[X1] a.s: (the strong version). The rst version was proved by J. Bernoulli in his Ars Conjectandi from around 1700: Xi took values 0 or 1 and convergence in weak sense (convergence in probability). The modern version is due to e¤orts of Chebyshev, Markov, Borel, Cantelli, and Kolmogorov. 1

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Ergodic theorems for noncommuting random products

Soit G un groupe continu et μ une probabilite a support compact de densite par rapport a la mesure de Haar. Soient X 1 , X 2 , … une marche aleatoire sur G associee a μ. On demontre que chaque fonction harmonique bornee est constante si et seulement si (1/n)|X n |→0 quand n→∞

G. Alexopoulos

Papers

On the mean distance of random walks on groups