Lévy flights in confining environments: Random paths and their statistics

Abstract

We analyze a specific class of random systems that are driven by a symmetric Lévy stable noise. In view of the Lévy 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) ρ * (x) ∼ exp[−Φ(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 ρ(x, t) dynamics directly from the random paths statistics. A priori given data are jump transition rates entering the master equation for ρ(x, t) and its target pdf ρ * (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 µ ∈ (0, 2).

Path-wise versus kinetic modeling for equilibrating non-Langevin jump-type processes

“…However, it is not at all obvious, albeit anticipated in Ref. [14], that the path-wise route gives rise to the same evolution of the probability distribution as that coming out from a direct integration of the master equation (1). A reliable validity test is necessary here.…”

“…[14] was to infer a direct non-Langevin path-wise description of random motion, on the sole basis of the pre-defined master equation. The task has been accomplished by means of a properly tailored Gillespie's algorithm.…”

“…Our focus is on the former issue, while the latter (efficiency) topic will be briefly commented. We quantify a pdf ρ( ) evolution, that is driven by a non -Langevin jump-type process on R, in terms of the master equation, [14][15][16]:…”

“…On the other hand, if one can devise a path-wise picture of the underlying random dynamics, whose statistical consequence is the evolution (1) Brought to you by | MIT Libraries Authenticated Download Date | 5/11/18 9:24 AM of ρ( ), we talk about an indirect path-wise method of solution of the master equation. The latter case was the subject of our recent paper [14], followed by a straightforward extension from 1D to 2D, [17]. Since generically there is no Langevin representation of the random dynamics in question, our main goal in Ref.…”

“…Since generically there is no Langevin representation of the random dynamics in question, our main goal in Ref. [14] has been to establish the appropriate path-wise description of the underlying jump-type process and subsequently to infer the ρ( ) dynamics directly from the random paths statistics data. However, it is not at all obvious, albeit anticipated in Ref.…”