Discretization of bounded harmonic functions on Riemannian manifolds and entropy

We give conditions under which the space of bounded harmonic functions on a Riemannian manifold M is naturally isomorphic to the space of bounded harmonic functions of a Markov chain on a discrete net X M arising from a discretization procedure for the pair (M; X). If, further, M is a regular covering manifold and the net is invariant with respect to the deck transformation group, then the entropy of the arising random walk on X equals the entropy of the Brownian motion on M times the average stopping time of the discretization procedure. During the last few years a lot of papers devoted to the discrete potential theory has appeared. It turns out that this theory is to a large extent parallel to the potential theory on Riemannian manifolds (see, e.g., a survey 1]). Thus one can naturally ask about any direct relationships between the potential theory on Riemannian manifolds and on graphs adapted in some way to the Riemannian structure. In probabilistic terms this is the question about a correspondence between the Brownian motion on a Riemannian manifold and appropriately taken Markov chain on a net in the manifold. The situation with the connection between the transience of the Brownian motion on a Riemannian manifold (i.e. existence of the Green function on this manifold) and the transience of a reversible Markov chain on a net in this manifold (hyperbolicity, in terminology of 13]) turned out to be quite simple. Namely, under natural cocompact-ness conditions the net is transient if and only if the manifold is transient itself 12]. The same is true about the isoperimetric properties of the manifold and its cocompact net 7]. In particular, the spectral radius of the random walk on a cocompact net is strictly less than 1 ii zero doesn't belong to the spectrum of the Laplacian on the manifold.