Levy flights in a steep potential well

Levy 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 Levy flights in potentials of the type U(x) proportional to \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 delta-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 Levy flight. These properties of Levy 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.