Lévy flights in inhomogeneous environments

Abstract

We study the long time asymptotics of probability density functions (pdfs) of Lévy flights in different confining potentials. For that we use two models: Langevin - driven and (Lévy - Schrödinger) semigroup - driven dynamics. It turns out that the semigroup modeling provides much stronger confining properties than the standard Langevin one. Since contractive semigroups set a link between Lévy flights and fractional (pseudo-differential) Hamiltonian systems, we can use the latter to control the long - time asymptotics of the pertinent pdfs. To do so, we need to impose suitable restrictions upon the Hamiltonian and its potential. That provides verifiable criteria for an invariant pdf to be actually an asymptotic pdf of the semigroup-driven jump-type process. For computational and visualization purposes our observations are exemplified for the Cauchy driver and its response to external polynomial potentials (referring to Lévy oscillators), with respect to both dynamical mechanisms.

Major revision

References in corpus (3)

• Fractional Quantum Mechanics
• Levy flights in confining potentials
• Intrinsic ultracontractivity for Schrodinger operators based on fractional Laplacians

Cited by in corpus (14)

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• Levy targeting and the principle of detailed balance
• Nonlocally-induced (fractional) bound states: Shape analysis in the infinite Cauchy well
• Brownian motion in trapping enclosures: Steep potential wells, bistable wells and false bistability of induced Feynman-Kac (well) potentials
• Heavy-tailed targets and (ab)normal asymptotics in diffusive motion
• Ultrarelativistic bound states in the spherical well
• Ultrarelativistic bound states in the shallow spherical well
• Nonlocally-induced (quasirelativistic) bound states: Harmonic confinement and the finite well
• Fractional Laplacians and Levy flights in bounded domains
• A maximal $L_p$-regularity theory to initial value problems with time measurable nonlocal operators generated by additive processes
• Thermalization of Levy flights: Path-wise picture in 2D