Levy targeting and the principle of detailed balance

Details Levy targeting and the principle of detailed balance Levy targeting and the principle of detailed balance

2011-05-07

arxiv.org/abs/1105.1435v1

Phys. Rev. E 84, (2011), 011109

Physics

Condensed Matter

Statistical Mechanics

Scientific paper

We investigate confined L\'{e}vy flights under premises of the principle of detailed balance. The master equation admits a transformation to L\'{e}vy - Schr\"{o}dinger 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 $0<\mu <2$ of L\'{e}vy drivers: given an invariant probability density function (pdf), specify the jump - type dynamics for which this pdf is a long-time asymptotic target. Our ("$\mu$-targeting") method is exemplified by Cauchy family and Gaussian target pdfs. We solve the reverse engineering problem for so-called L\'{e}vy oscillators: given a quadratic semigroup potential, find an asymptotic pdf for the associated master equation for arbitrary $\mu$.

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