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This paper presents necessary and sufficient conditions for on- andoff-diagonal transition probability estimates for random walks on weightedgraphs. On the integer lattice and on may fractal type graphs both the volumeof 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 andtwo-sided estimate is given if uniformity in the space assumed only for themean exit time.

The volume and time comparison principle and transition probability estimates for random walks