The volume and time comparison principle and transition probability   estimates for random walks
  This paper presents necessary and sufficient conditions for on- and off-diagonal transition probability estimates for random walks on weighted graphs. On the integer lattice and on may fractal type graphs both the volume of a ball and the mean exit time from a ball is independent of the centre, uniform in space. Here the upper estimate is given without such restriction and two-sided estimate is given if uniformity in the space assumed only for the mean exit time. 
Andras Telcs