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Levy targeting and the principle of detailed balance

We investigate confined Lévy flights under premises of the principle of detailed balance. The master equation admits a transformation to Lévy - Schrödinger semigroup dynamics (akin to a mapping of the Fokker-Planck equation into the generalized diffusion equation). We solve a stochastic targeting problem for arbitrary stability index $0<μ<2$ of Lévy drivers: given an invariant probability density function (pdf), specify the jump - type dynamics for which this pdf is a long-time asymptotic target. Our ("$μ$-targeting") method is exemplified by Cauchy family and Gaussian target pdfs. We solve the reverse engineering problem for so-called Lévy oscillators: given a quadratic semigroup potential, find an asymptotic pdf for the associated master equation for arbitrary $μ$.