A recent theorem by M. Manstavicius (2004) provided a link between a certain function of transition probabilities of a strong Markov process and the boundedness of the $p$-variation of its trajectories. Here one assumption of that theorem is relaxed and an example is constructed to show that the Markov property cannot be easily dispensed with.
"A Non-Markovian Process with Unbounded $p$-Variation." Electron. Commun. Probab. 10 17 - 28, 2005. https://doi.org/10.1214/ECP.v10-1128