Estimates for differences and Harnack inequality for difference operators coming from random walks with symmetric, spatially inhomogeneous, increments

Difference operators arising from random walks with symmetric increments are studied. If the random walk is spatially homogeneous, then estimates of the first and second differences of harmonic functions are given and a Harnack inequality is proved. In the spatially inhomogeneous case, a Harnack inequality for superharmonic functions is proved, giving a discrete version of a result of Krylov and Safonov. This is used to give an estimate for differences of harmonic functions and applied to show existence of harmonic measure for spatially inhomogeneous walks. 1