From Ornstein-Uhlenbeck dynamics to long-memory processes and fractional Brownian motion

This article establishes a natural physical path leading from "regular" Ornstein-Uhlenbeck dynamics to "anomalous" long-memory processes and, thereafter, to fractional Brownian motion. Considering a system composed of n different parts-each part conducting its own Ornstein-Uhlenbeck dynamics, and all parts being perturbed by a common external Lévy noise-we show that the collective system-dynamics, in the limit n-->infinity , converges to a temporal moving-average of the driving noise. The limiting moving-average process, in turn, can possess a long memory-in which case, when observed over large time scales, further yields fractional Brownian motion. The temporal correlation structure of the limiting moving-average process turns out to be determined by the structural statistical variability of the system's composing parts. Thus, the emergence of a long memory is a consequence of the intrinsic "quenched disorder" present at the system's formation epoch rather than the consequence of the external annealed disorder carried in continuously by the driving noise.