A Gaussian upper bound for the iterated kernels of Markov chains is obtained under some natural conditions. This result applies in particular to simple random walks on any locally compact unimodular group $G$ which is compactly generated. Moreover, if $G$ has polynomial volume growth, the Gaussian upper bound can be complemented with a similar lower bound. Various applications are presented. In the process, we offer a new proof of Varopoulos' results relating the uniform decay of convolution powers to the volume growth of $G$.
"Gaussian Estimates for Markov Chains and Random Walks on Groups." Ann. Probab. 21 (2) 673 - 709, April, 1993. https://doi.org/10.1214/aop/1176989263