We prove that a two sided sub-Gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.

this paper is to estimate the hitting probability function / K (t; x) := P x (9 s 2 [0; t] ; X s 2 K) where K ae M is a fixed compact set. In words, / K (t; x) is the probability that Brownian motion started at x hits K by time t. Our goal is to obtain precise estimates on / K for all t ? 0 and x outside a neighborhood of K, hence avoiding the somewhat different question of the behavior of / K near the boundary of K. In the context of Riemannian manifolds, this natural question has been considered only in a handful of papers including [2], [4]. We were led to study / K in our attempt to develop sharp heat kernel estimates on manifolds with more than one end. Indeed, the proof of the heat kernel estimates announced in [20] depends in a crucial way on the results of the present paper (see [21]). In this context, it turns out to be important to estimate also the time derivative @ t / K (t; x) which is a positive function. We develop a general approach which allows to obtain estimates of / K in terms of the heat kernel p(t; x; y) or closely related objects such as the Dirichlet heat kernel p U (t; x; y) of some open set U . In the case when X t is transient, that is, M is non-parabolic, we show that the behavior of / K (t; x), away from K, is comparable to that of Z t 0 p(s; x; y)ds; where y is a reference point on @K. If (X t ) t?0 is recurrent, that is, M is parabolic, we obtain similar estimates through Z t 0 p U (s; x; y)ds where U is a certain region slightly larger than\Omega := M n K. We also show that @ t / K (t; x) is comparable to p\Omega (t; x; y) where y stays at a certain distance from @K. For precise statements, see Theorems 3.3, 3.5, 3.7 and Corollaries 3.9, 3.10. Using the known results concerning the heat kernel p(t; x; y) and the results of [23...