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

Abstract

We investigate the non-Langevin relative of the Lévy-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 Lévy flights in superharmonic potential wells, with a focus on the extremally steep well regime, and address the issue of its (spectral) "closeness" to the Lévy jumptype 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 Lévy 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.