Levy flights in confining environments: Random paths and their statistics

We analyze a specific class of random systems that are driven by a symmetric L'{e}vy stable noise. In view of the L'{e}vy noise sensitivity to the confining “potential landscape” where jumps take place (in other words, to environmental inhomogeneities), the pertinent random motion asymptotically sets down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) $\rho_(x) \sim \exp [-\Phi (x)]$. Since there is no Langevin representation of the dynamics in question, our main goal here is to establish the appropriate path-wise description of the underlying jump-type process and next infer the $\rho (x,t)$ dynamics directly from the random paths statistics. A priori given data are jump transition rates entering the master equation for $\rho (x,t)$ and its target pdf $\rho_(x)$. We use numerical methods and construct a suitable modification of the Gillespie algorithm, originally invented in the chemical kinetics context. The generated sample trajectories show up a qualitative typicality, e.g. they display structural features of jumping paths (predominance of small vs large jumps) specific to particular stability indices $\mu \in (0,2)$.

💡 Research Summary

The paper addresses stochastic processes driven by symmetric Lévy‑stable noise that evolve under the influence of a confining “potential landscape” Φ(x). Because Lévy noise generates non‑local jumps, a conventional Langevin description is unavailable. Instead, the authors start from a master (or transport) equation for the probability density ρ(x,t):

∂ₜρ(x)=∫