We make a connection between the continuous time and lazy discrete time Markov chains through the comparison of cutoffs and mixing time in total variation distance. For illustration, we consider finite birth and death chains and provide a criterion on cutoffs using eigenvalues of the transition matrix.
"Comparison of cutoffs between lazy walks and Markovian semigroups." J. Appl. Probab. 50 (4) 943 - 959, December 2013. https://doi.org/10.1239/jap/1389370092