Anomalous diffusion: temporal non-Markovianity and weak ergodicity breaking

Traditionally, the discrimination between a Markovian and a non-Markovian process is based on the definition. If the process is Markovian, its transition probability does not depend on the history of the process and it fulfills the Smoluchowski–Chapman–Kolmogorov equation. A practical verification of these two criteria is not always possible or fully conclusive. Therefore, we present an additional method which can be used to confirm the simplest version of Markovianity. This method is based on the properties of sums of independent random variables. We apply the presented method to prove the increment dependent character of an anomalous process combining long waiting times with long jumps. Such a process, despite being non-Markovian in nature, due to a competition between long waiting times and long jumps, can reveal 'normal' behavior. We also demonstrate that this anomalous process breaks the ergodicity in the weak sense. Finally, we apply the suggested method to some experimental time series proving their Markovian nature for small timescales