On positively divisible non-Markovian processes

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.

arXiv comment14 pages, 1 figure

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Cited by in corpus (2)

• Strong symmetries in collision models and physical dilations of covariant quantum maps
• Unification of stochastic matrices and quantum operations for N-level systems