A Discrete Time Harnack Inequality and Its Applications

Furstenberg in his paper Fu71] considered the Brownian motion on a Rie-mannian symmetric space S and proved that if G is a lattice in the corresponding semi-simple Lie group, then for any point o 2 S there exists a probability measure on G such that the harmonic measure o of the point o on the Furstenberg boundary of the space S is-stationary, i.e., o = P g (g)g o. It implies that for any bounded harmonic function on S its restriction to the G-orbit of o is a-harmonic function. In fact, his argument required just recurrence of the Brownian motion on the quotient space S=G. Lyons and Sullivan LS84] noticed that Furstenberg's construction is applicable to the Brownian motion on an arbitrary covering Riemannian manifold f M provided the quotient manifold M is recurrent (actually, they showed how it can be done even without any homogeneity assumptions) and proved that the resulting random walk on the deck transformations group G well approximates the original Brownian motion in various senses. As it was later shown by the author in Ka92a], this random walk also has the same bounded harmonic functions as the original Brownian motion: if f is a bounded harmonic function of the Laplace-Beltrami operator on f M, then its restriction to the orbit Go = G is-harmonic, every bounded-harmonic function can be uniquely obtained in this way, and this correspondence is an isometry between the spaces of bounded harmonic functions on f M and G. On a measure-theoretical level it just means that the corresponding Poisson boundaries are isomorphic as measure spaces Ka92b]. The main idea of this discretization procedure is to use the Harnack inequality. In the Riemannian case it almost automatically follows from bound-edness of geometry of f M and continuity of sample paths of the Brownian motion. In order to obtain a Harnack inequality for discrete time Markov operators one has to impose on the operator some locality and \bounded geometry" assumptions. We establish a Harnack inequality (Theorem 1) for discrete time Markov operators on a general metric space (X; d) under several additional conditions (P 1) { (P 5), the most important of which are