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We study the Poisson boundary (≡ representation of bounded harmonic functions) of Markov operators on discrete state spaces that are invariant under the action of a transitive group of permutations. This automorphism group is locally compact, but not necessarily discrete or unimodular. The main technical tool is the entropy theory which we develop along the same lines as in the case of random walks on countable groups, while, however, the implementation is different and exploits discreteness of the state space on the one hand and the path space of the induced random walk on the nondiscrete group on the other. Various new examples are given as applications, including a description of the Poisson boundary for random walks on vertex-transitive graphs with infinitely many ends and on the Diestel-Leader graphs. 1. Introduction. The Poisson boundary of a Markov operator P is defined as the space of ergodic components of the time shift in the path space. Via the Poisson formula the space of bounded P -harmonic functions is isometric to the space of bounded measurable functions on the Poisson boundary. This characterizes the Poisson boundary up to a measure-theoretic isomorphism. It also has a topological description in terms of the Martin boundary, where it consists of the set of possible limits of the Markov chain at the boundary together with the family of corresponding harmonic hitting (limit) distributions; see, for example, (14). However, we emphasize that the Poisson boundary is primarily a measure-theoretical notion and all the objects connected with the Poisson boundary are defined modulo subsets of measure 0 (see (24) for more details). The principal purpose of this paper is to present methods for determining the Poisson boundary in the case when P is a (space) homogeneous Markov operator, that is, when there is a locally compact group G of permutations of X that acts transitively on X and leaves P invariant: p(gx, gy) = p(x, y) ∀x, y ∈ X, g ∈ G. (In this paper we always deal with Markov chains which are time homogeneous in the traditional sense; that is, their transition probabilities do not depend on time. "Homogeneous" will always mean space homogeneous.) This comprises the case of random walks on countable groups, where the Poisson boundary has been studied extensively by Kaimanovich and Vershik (29), Ledrappier (32) and

Boundary and Entropy of Space Homogeneous Markov Chains