On Jump-Diffusive Driving Noise Sources: Some Explicit Results and Applications

TL;DR

This paper explores linear and nonlinear shot noise models driven by compound Poisson processes with Erlang-distributed jumps, deriving explicit PDE forms and analyzing collective dynamics and traveling waves in large populations.

Contribution

It provides explicit PDE forms for shot noise models with state-dependent rates and characterizes collective behaviors, including wave speeds influenced by jump size.

Findings

Master equation reduces to a spatial m-th order PDE

Traveling wave speed can be controlled by jump size m

Explicit solutions for certain nonlinear SDEs with shot noise

Abstract

We study some linear and nonlinear shot noise models where the jumps are drawn from a compound Poisson process with jump sizes following an Erlang-$m$ distribution. We show that the associated Master equation can be written as a spatial $m^{{\rm th}}$ order partial differential equation without integral term. This differential form is valid for state-dependent Poisson rates and we use it to characterize, via a mean-field approach, the collective dynamics of a large population of pure jump processes interacting via their Poisson rates. We explicitly show that for an appropriate class of interactions, the speed of a tight collective traveling wave behavior can be triggered by the jump size parameter $m$. As a second application we consider an exceptional class of stochastic differential equations with nonlinear drift, Poisson shot noise and an additional White Gaussian Noise term, for which explicit solutions to the associated Master equation are derived.