Markov-breaking and the emergence of long memory in Ornstein-Uhlenbeck systems

We consider a complex system composed of many non-identical parts where (i) the dynamics of each part are Ornstein-Uhlenbeck; (ii) all parts are driven by a common external Lévy noise; and (iii) the system's collective output is the averaged aggregate of the outputs of its parts. Whereas the dynamics on the 'microscopic' parts-level are Markov, the dynamics on the 'macroscopic' system-level are not Markov - and may display a long memory. Moreover, the universal temporal scaling limit of the system's output, in the presence of long memory, is fractional Brownian motion. The model presented is analytically tractable, and gives closed-form quantitative characterizations of both the Markov-breaking phenomenon and the emergence of long memory.