Nonlinear Shot Noise: From aggregate dynamics to maximal dynamics

¹  Department of Technology Management, Holon Institute of Technology - Holon 58102, Israel
²  School of Chemistry, Sackler Faculty of Exact Sciences, Tel Aviv University - Tel Aviv 69978, Israel

Corresponding authors: eliazar@post.tau.ac.il klafter@post.tau.ac.il

Received: 1 January 2007
Accepted: 3 April 2007

Abstract

We consider Nonlinear Shot Noise systems in which external shots hit the system following an arbitrary Poissonian inflow and, after impact, dissipate to zero governed by an arbitrary nonlinear decay mechanism. The “standard” Shot Noise process tracks the shots aggregate (at any given time) and is non-Markov. In this letter we shift from aggregate dynamics to maximal dynamics —tracking the magnitude of the largest shot present in the system (at any given time). This yields a class of stochastic processes which: i) display a wide spectrum of random decay-surge evolutionary patterns; ii) are intrinsically nonlinear in both their decay and surge mechanisms; but yet, iii) turn out to be Markovian and analytically tractable. A detailed quantitative statistical analysis of this class of processes is presented.