Lévy flights in a steep potential well

Aleksei V. Chechkin*, Vsevolod Yu Gonchar, Joseph Klafter, Ralf Metzler, Leonid V. Tanatarov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

129 Scopus citations


Lévy flights in steeper than harmonic potentials have been shown to exhibit finite variance and a critical time at which a bifurcation from an initial monomodal to a terminal bimodal distribution occurs (Chechkin et al., Phys. Rev. E 67:010102(R) (2003)). In this paper, we present a detailed study of Lévy flights in potentials of the type U(x) α |x|c with c > 2. Apart from the bifurcation into bimodality, we find the interesting result that for c > 4 a trimodal transient exists due to the temporal overlap between the decay of the central peak around the initial δ-condition and the building up of the two emerging side-peaks, which are characteristic for the stationary state. Thus, for certain system parameters there exists a transient trimodal distribution of the Lévy flight. These properties of Lévy flights in external potentials of the power-law type can be represented by certain phase diagrams. We also present details about the proof of multimodality and the numerical procedures to establish the probability distribution of the process.

Original languageEnglish
Pages (from-to)1505-1535
Number of pages31
JournalJournal of Statistical Physics
Issue number5-6
StatePublished - Jun 2004


  • Classical transport
  • Random walks and Lévy flights
  • Stochastic analysis methods (Fokker-Planck, Langevin, etc.)
  • Stochastic processes


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