On Stochastic Processes Driven by Ballistic Noise Sources

Among the rich collection of noise sources that can be used to drive stochastic differential equations (SDE), we here focus on Markov processes that themselves are functions of the standard Brownian motion and of the Telegrapher’s process. Our present noise sources can be viewed as lumped versions of Markov processes with enlarged states spaces. The resulting Markov processes exhibit ballistic non-Gaussian noise features with long time range correlations. When used as simple additive noise sources in SDEs, these ballistic fluctuations may generate noise induced structures (i.e., noise induced transitions), a behavior that is usually only observed for multiplicative de-correlated noise sources. Our ballistic noise sources are then used in a selection of applications taken from control theory and multi-agents systems for which exact analytic results are derived.