Nonlinear shot noise, memory systems, and all-time hit parades

Consider the evolution of a memory system ‘fed’ by an external event-process. New memories are continuously recorded by the system. Simultaneously, the recollection of old memories continuously fades away. Thus, at a given time epoch the memory system ranks all past events according to present importance-magnitudes attributed to them. Illustratively, the memory system is an all-time hit parade run continuously in time. Motivated by a recently-introduced nonlinear shot noise system-model [I. Eliazar, J. Klafter, Physica A, in press (titled: non-linear shot noise: Lévy, Noah, & Joseph).], we explore a memory system-model in which: (i) the external events follow an arbitrary time-homogeneous Poisson point process; and (ii) the ‘fading’ of memories is governed by an arbitrary nonlinear differential-equation dynamics. A Poissonian analysis of the model is conducted, addressing questions such as: How do memories get constructed and degraded? How does the memory process evolve? What is its stationary structure? What is its correlation structure? In addition, a Poissonian eigenvalue problem, arising in this context, is studied.