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 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 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.

arXiv comment25 pp, 7 Figs, new references added

References in corpus (21)

• Ten equivalent definitions of the fractional Laplace operator
• Levy Flights in External Force Fields: Langevin and Fractional Fokker-Planck Equations, and their Solutions
• Levy--Brownian motion on finite intervals: Mean first passage time analysis
• Lévy flights versus Lévy walks in bounded domains
• Ornstein-Uhlenbeck-Cauchy Process
• Fractional Brownian motion in a finite interval: correlations effect depletion or accretion zones of particles near boundaries
• Fractional Brownian motion with a reflecting wall
• Steady-State Lévy Flights in a Confined Domain
• Probability density of the fractional Langevin equation with reflecting walls
• Feynman-Kac Kernels in Markovian Representations of the Schroedinger Interpolating Dynamics
• Levy flights in confining potentials
• Killing (absorption) versus survival in random motion
• Blow-up phenomena in nonlocal eigenvalue problems: when theories of $L^1$ and $L^2$ meet
• Transition in the decay rates of stationary distributions of Lévy motion in an energy landscape
• Levy targeting and the principle of detailed balance
• Nonlocally-induced (fractional) bound states: Shape analysis in the infinite Cauchy well
• Multifractal properties of sample paths of ground state-transformed jump processes
• Ultrarelativistic (Cauchy) spectral problem in the infinite well
• Fractional calculus for power functions
• Levy flights in confining environments: Random paths and their statistics
• Singular boundary behaviour and large solutions for fractional elliptic equations

Cited by in corpus (3)

• Brownian motion in trapping enclosures: Steep potential wells, bistable wells and false bistability of induced Feynman-Kac (well) potentials
• Levy processes in bounded domains: Path-wise reflection scenarios and signatures of confinement
• How to simulate Lévy flights in a steep potential: An explicit splitting numerical scheme