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

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.