Discretization of bounded harmonic functions on Riemannian manifolds and entropy
Vadim A. Kaimanovich · 1992
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 ae 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. 1980 Mathematics Subject Classification (1985 Revision): 31C12, 58G32, 60J50. 0. Introduction 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 graph...