Lévy flights in confining environments: Random paths and their statistics

Abstract

We analyze a specific class of random systems that, while being driven by a symmetric Lévy stable noise, asymptotically set down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) ρ∗(x)∼exp[−Φ(x)]. This behavior needs to be contrasted with the standard Langevin representation of Lévy jump-type processes. It is known that the choice of the drift function in the Newtonian form ∼−∇Φ excludes the existence of the Boltzmannian pdf ∼exp[−Φ(x)] (Eliazar–Klafter no go theorem). In view of this incompatibility statement, our main goal here is to establish the appropriate path-wise description of the equilibrating jump-type process. A priori given inputs are (i) jump transition rates entering the master equation for ρ(x,t) and (ii) the target (invariant) pdf ρ∗(x) of that equation, in the Boltzmannian form. We resort to 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 μ∈(0,2). The obtained random paths statistical data allow us to infer an associated pdf ρ(x,t) dynamics which stands for a validity check of our procedure. The considered exemplary Boltzmannian equilibria ∼exp[−Φ(x)] refer to (i) harmonic potential Φ∼x2, (ii) logarithmic potential Φ∼nln(1+x2) with n=1,2 and (iii) locally periodic confining potential Φ∼sin2(2πx),|x|≤2, Φ∼(x2−4),|x|>2.