title: Lévy targeting and the principle of detailed balance.
abstract: We investigate confining mechanisms for Lévy flights under premises of the principle of detailed balance. In this case, the master equation of the jump-type process admits a transformation to the Lévy-Schrödinger semigroup dynamics akin to a mapping of the Fokker-Planck equation into the generalized diffusion equation. This sets a correspondence between above two stochastic dynamical systems, within which we address a (stochastic) targeting problem for an arbitrary stability index μ ε (0,2) of symmetric Lévy drivers. Namely, given a probability density function, specify the semigroup potential, and thence the jump-type dynamics for which this PDF is actually a long-time asymptotic (target) solution of the master equation. Here, an asymptotic behavior of different μ-motion scenarios ceases to depend on μ. That is exemplified by considering Gaussian and Cauchy family target PDFs. A complementary problem of the reverse engineering is analyzed: given a priori a semigroup potential, quantify how sensitive upon the choice of the μ driver is an asymptotic behavior of solutions of the associated master equation and thus an invariant PDF itself. This task is accomplished for so-called μ family of Lévy oscillators.
authors.affiliation: Institute of Physics, University of Opole, 45-052 Opole, Poland.
authors.forename: Piotr
authors.initials: P
authors.lastname: Garbaczewski
authors.forename: Vladimir
authors.initials: V
authors.lastname: Stephanovich
mesh_terms: D000465:Algorithms; D004058:Diffusion; D008390:Markov Chains; D015233:Models, Statistical; D009068:Movement; D016011:Normal Distribution; D009991:Oscillometry; D010825:Physics / Q000379:methods*; D013269:Stochastic Processes; D013816:Thermodynamics