The discrete integral maximum principle and its applications

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Gaussian Upper Bounds for Heat Kernels of Continuous Time Simple Random Walks (2011, methodology): “For the discrete time simple random walk on Cp,∞(ω), Gaussian upper bounds are obtained in [7] as an application of their discrete time heat kernel estimates.”
Li-Yau inequality on graphs (2013, background): “See also [2, 7, 8, 9] for other attempts to avoid or partially bypass the chain rule in the discrete setting.”
Gaussian heat kernel upper bounds via the Phragmén–Lindelöf theorem (2006, unknown): “A discrete time version of the Davies–Gaffney estimate was discussed in [24]; however, it is not clear how to extend the methods of the present work to the discrete time case, in order to replace the use of the rather technical discrete integrated maximum principle as in [24].”