Geodesic
The volume and time comparison principle and transition probability estimates for random walks
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03), DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03) (2003).

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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 are independent of the center, uniform in space. Here the upper estimate is given without such restriction and two-sided estimate is given if the mean exit time is independent of the center but the volume is not. 
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     author = {Telcs, Andr\'as},
     title = {The volume and time comparison principle and transition probability estimates for random walks},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03)},
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     doi = {10.46298/dmtcs.3334},
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     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3334/}
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Telcs, András. The volume and time comparison principle and transition probability estimates for random walks. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03), DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03) (2003). doi : 10.46298/dmtcs.3334. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3334/

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