On positively divisible non-Markovian processes

2024/01/23 by Canturk, Bilal, Breuer, Heinz-Peter · 1 citation
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2401.12715

Abstract

There are some positively divisible non-Markovian processes whose transition matrices satisfy the Chapman-Kolmogorov equation. These processes should also satisfy the Kolmogorov consistency conditions, an essential requirement for a process to be classified as a stochastic process. Combining the Kolmogorov consistency conditions with the Chapman-Kolmogorov equation, we derive a necessary condition for positively divisible stochastic processes on a finite sample space. This necessary condition enables a systematic approach to the manipulation of certain Markov processes in order to obtain a positively divisible non-Markovian process. We illustrate this idea by an example and, in addition, analyze a classic example given by Feller in the light of our approach.

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