Trajectory Statistics of Confined L\'evy Flights and Boltzmann-type Equilibria

Abstract

We analyze a specific class of random systems that are driven by a symmetric Lévy stable noise, where the Langevin representation is absent. In view of the Lévy noise sensitivity to environmental inhomogeneities, the pertinent random motion asymptotically sets down at the Boltzmann-type equilibrium, represented by a probability density function (pdfHere, we infer pdf ρ(x, t) based on numerical path-wise simulation of the underlying jump-type process. A priori given data are jump transition rates entering the master equation for ρ(x, t) and its target pdf ρ * (x). To simulate the above processes, we construct a suitable modification of the Gillespie algorithm, originally invented in the chemical kinetics context. We exemplified our algorithm simulating different jump-type processes and discuss the dynamics of real physical systems, where it can be useful.