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

Iddo Eliazar*, Joseph Klafter

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Article number122001
JournalJournal of Physics A: Mathematical and Theoretical
Issue number12
StatePublished - 28 Mar 2008


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