Lévy flights in confining environments: Random paths and their statistics

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 ∼ x 2 , (ii) logarithmic potential Φ ∼ n ln ( 1 + x 2 ) with n = 1 , 2 and (iii) locally periodic confining potential Φ ∼ sin 2 ( 2 π x ) , | x | ≤ 2 , Φ ∼ ( x 2 − 4 ) , | x | > 2 .