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Trajectory statistics of confined Lévy flights and Boltzmann-type equilibria. (English) Zbl 1371.60080
Summary: 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 (pdf) $$\rho_*(x)\sim\exp[-\Phi(x)]$$. Here, we infer pdf $$\rho(x,t)$$ 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 $$\rho(x,t)$$ and its target pdf $$\rho_*(x)$$. 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.
##### MSC:
 60G51 Processes with independent increments; Lévy processes 82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
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