(with A. Erschler)

We give a construction of homomorphisms from a group G into R using random walks on G. The construction is an alternative to [KL07] that works in more general situations. Applications include an estimate on the drift of random walk on groups of subexponential group admitting no nontrivial homomorphism to Z and inequalities between the asymptotic drift L(n) and the asymptotic entropy H(n). Some of the entropy estimates obtained have applications independent of the homomorphism construction, for example a Liouville-type theorem for slowly growing harmonic functions on groups of subexponential growth and on some groups of exponential growth.

(with F. Ledrappier)

Let M be a complete connected Riemannian manifold with bounded sectional curvature. Under the assumption that M is a regular covering of a manifold with finite volume, we establish that M is Liouville if, and only if, the average rate of escape of Brownian motion on M is sublinear.

(with F. Ledrappier)

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.

(with W. Woess)

Let T_q be the homogeneous tree with degree q + 1 = 3 and F a finitely generated group whose Cayley graph is T_q. The associated lamplighter group is the wreath product L with F , where L is a finite group. For a large class of random walks on this group, we prove almost sure convergence to a natural geometric boundary. If the probability law governing the random walk has finite first moment, then the probability space formed by this geometric boundary together with the limit distribution of the random walk is proved to be maximal, that is, the Poisson boundary. We also prove that the Dirichlet problem at infinity is solvable for continuous functions on the active part of the boundary, if the lamplighter operates at bounded range.

(with F. Ledrappier)

We prove a general noncommutative law of large numbers. This applies in particular to random walks on any locally finite homogeneous graph, as well as to Brownian motion on Riemannian manifolds which admit a compact quotient. It also generalizes Oseledec’s multiplicative ergodic theorem. In addition, we show that epsilon-shadows of any ballistic random walk with finite moment on any group eventually intersect. Some related results concerning Coxeter groups and mapping class groups are recorded in the last section.

Discrete Mathematics and Theoretical Computer Science Proceedings AC, (2003) pp. 137-144

We obtain a new result concerning harmonic functions on infinite Cayley graphs X: either every nonconstant harmonic function has infinite radial variation in a certain uniform sense, or there is a nontrivial boundary with hyperbolic properties at infinity of X. In the latter case, relying on a theorem of Woess, it follows that the Dirichlet problem is solvable with respect to this boundary. Certain relations to group cohomology are also discussed.

We prove that almost every path of a random walk on a finitely generated nonamenable group converges in the compactification of the group introduced by W. J. Floyd. In fact, we consider the more general setting of ergodic cocycles of some semigroup of one-Lipschitz maps of a complete metric space with a boundary constructed following Gromov. We obtain in addition that when the Floyd boundary of a finitely generated group is non-trivial, then it is in fact maximal in the sense that it can be identified with the Poisson boundary of the group with reasonable measures. The proof relies on works of Kaimanovich together with visibility properties of Floyd boundaries. Furthermore, we discuss mean proximality and a conjecture of McMullen. Lastly, related statements about the convergence of certain sequences of points, for example quasigeodesic rays or orbits of one-Lipschitz maps, are obtained.

(Ed. by V.A. Kaimanovich, in collab. with K. Schmidt, W. Woess) de Gruyter, 2004.

This paper describes some situations when random walks (or related processes) of linear rate of escape converge in direction in various senses. We discuss random walks on isometry groups of fairly general metric spaces, and more specifically, random walks on isometry groups of nonpositive curvature, isometry groups of reflexive Banach spaces, and linear groups preserving a proper cone. We give an alternative proof of the main tool from subadditive ergodic theory and make a conjecture in this context involving Busemann functions.

(with G. Margulis)

We study integrable cocycles u(n,x) over an ergodic measure preserving transformation that take values in a semigroup of nonexpanding maps of a nonpositively curved space Y, e.g. a Cartan-Hadamard space or a uniformly convex Banach space. It is proved that for any y]Y and almost all x, there exist AS 0 and a unique geodesic ray n (t,x) in Y starting at y such that\begin{equation*} \lim_{n\rightarrow \infty }\frac 1nd(\gamma (An,x),u(n,x)y)=0. \end{equation*} In the case where Y is the symmetric space GLN(Â)/ON(Â) and the cocycles take values in GLN(Â), this is equivalent to the multiplicative ergodic theorem of Oseledec. Two applications are also described. The first concerns the determination of Poisson boundaries and the second concerns Hilbert-Schmidt operators.