Levy-driven Langevin systems: Targeted stochasticity

Langevin dynamics driven by random Wiener noise ("white noise''), and the resulting Fokker-Planck equation and Boltzmann equilibria are fundamental to the understanding of transport and relaxation. However, there is experimental and theoretical evidence that the use of the Gaussian Wiener noise as an underlying source of randomness in continuous time systems may not always be appropriate or justified. Rather, models incorporating general Levy noises, should be adopted. In this work we study Langevin systems driven by general Levy, rather than Wiener, noises. Various issues are addressed, including: (i) the evolution of the probability density function of the system's state; (ii) the system's steady state behavior; and, (iii) the attainability of equilibria of the Boltzmann type. Moreover, the issue of reverse engineering is introduced and investigated. Namely: how to design a Langevin system, subject to a given Levy noise, that would yield a pre-specified "target'' steady state behavior. Results are complemented with a multitude of examples of Levy driven Langevin systems.