Levy flights in confining potentials

TL;DR

This paper investigates how Le9vy flights behave in confining potentials, revealing conditions under which their variance exists and stationary distributions form, with implications for modeling superdiffusive phenomena in complex systems.

Contribution

It demonstrates that different classes of Le9vy-driven processes can share stationary distributions despite different dynamics, extending the reverse engineering approach for designing targeted stochastic processes.

Findings

Both Langevin-driven and topological Le9vy processes can share stationary distributions.

Confining potentials can induce finite variance in Le9vy flights.

The results apply to diverse systems like polymers, geophysical flows, and climate models.

Abstract

We analyze confining mechanisms for L\'{e}vy flights. When they evolve in suitable external potentials their variance may exist and show signatures of a superdiffusive transport. Two classes of stochastic jump - type processes are considered: those driven by Langevin equation with L\'{e}vy noise and those, named by us topological L\'{e}vy processes (occurring in systems with topological complexity like folded polymers or complex networks and generically in inhomogeneous media), whose Langevin representation is unknown and possibly nonexistent. Our major finding is that both above classes of processes stay in affinity and may share common stationary (eventually asymptotic) probability density, even if their detailed dynamical behavior look different. That generalizes and offers new solutions to a reverse engineering (e.g. targeted stochasticity) problem due to I. Eliazar and J. Klafter [J. Stat. Phys. 111, 739, (2003)]: design a L\'{e}vy process whose target pdf equals a priori preselected one. Our observations extend to a broad class of L\'{e}vy noise driven processes, like e.g. superdiffusion on folded polymers, geophysical flows and even climatic changes.