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

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.