Levy flights in steep potential wells: Langevin modeling versus direct response to energy landscapes

TL;DR

This paper compares Langevin and non-Langevin models of Levy flights in steep potential wells, focusing on their steady-state behaviors and boundary conditions, highlighting differences in boundary reflection implementations.

Contribution

It introduces a non-Langevin model with direct response to energy landscapes and discusses boundary condition ambiguities for Levy processes in confined domains.

Findings

Non-Langevin model responds directly to potential landscapes.

Boundary reflection in Levy processes remains conceptually unresolved.

Steep potential wells influence Levy flight confinement behavior.

Abstract

We investigate the non-Langevin relative of the L\'{e}vy-driven Langevin random system, under an assumption that both systems share a common (asymptotic, stationary, steady-state) target pdf. The relaxation to equilibrium in the fractional Langevin-Fokker-Planck scenario results from an impact of confining conservative force fields on the random motion. A non-Langevin alternative has a built-in direct response of jump intensities to energy (potential) landscapes in which the process takes place. We revisit the problem of L\'{e}vy flights in superharmonic potential wells, with a focus on the extremally steep well regime, and address the issue of its (spectral) "closeness" to the L\'{e}vy jump-type process confined in a finite enclosure with impenetrable (in particular reflecting) boundaries. The pertinent random system "in a box/interval" is expected to have a fractional Laplacian with suitable boundary conditions as a legitimate motion generator. The problem is, that in contrast to amply studied Dirichlet boundary problems, a concept of reflecting boundary conditions and the path-wise implementation of the pertinent random process in the vicinity of (or sharply at) reflecting boundaries are not unequivocally settled for L\'{e}vy processes. This ambiguity extends to fractional motion generators, for which nonlocal analogs of Neumann conditions are not associated with path-wise reflection scenarios at the boundary, respecting the impenetrability assumption.