Random walk on graphs with regular resistance and volume growth

Abstract

In this paper characterizations of graphs satisfying heat kernel estimates for a wide class of space-time scaling functions are given. The equivalence of the two-sided heat kernel estimate and the parabolic Harnack inequality is also shown via the equivalence of the upper (lower) heat kernel estimate to the parabolic mean value (and super mean value) inequality. © Association des Publications de l'Institut Henri Poincaré, 2008.