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Lévy flights in confining environments: random paths and their statistics. (English) Zbl 1395.82089
Summary: 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) $$\rho_\ast(x) \sim \exp [- \Phi(x)]$$. 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 $$\sim - \nabla \Phi$$ excludes the existence of the Boltzmannian pdf $$\sim \exp [- \Phi(x)]$$ (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 $$\rho(x, t)$$ and (ii) the target (invariant) pdf $$\rho_\ast(x)$$ 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 $$\mu \in(0, 2)$$. The obtained random paths statistical data allow us to infer an associated pdf $$\rho(x, t)$$ dynamics which stands for a validity check of our procedure. The considered exemplary Boltzmannian equilibria $$\sim \exp [- \Phi(x)]$$ refer to (i) harmonic potential $$\Phi \sim x^2$$, (ii) logarithmic potential $$\Phi \sim n \ln(1 + x^2)$$ with $$n = 1, 2$$ and (iii) locally periodic confining potential $$\Phi \sim \sin^2(2 \pi x), | x | \leq 2$$, $$\Phi \sim(x^2 - 4), | x | > 2$$.
##### MSC:
 82B31 Stochastic methods applied to problems in equilibrium statistical mechanics 82B80 Numerical methods in equilibrium statistical mechanics (MSC2010)
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