Garbaczewski Piotr, Stephanovich Vladimir
Institute of Physics, University of Opole, 45-052 Opole, Poland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Jul;84(1 Pt 1):011142. doi: 10.1103/PhysRevE.84.011142. Epub 2011 Jul 27.
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.
我们在细致平衡原理的前提下研究了 Lévy 飞行的限制机制。在这种情况下，跳跃型过程的主方程允许变换为类似于福克 - 普朗克方程到广义扩散方程映射的 Lévy - 薛定谔半群动力学。这在上述两个随机动力系统之间建立了对应关系，在此对应关系中，我们针对对称 Lévy 驱动的任意稳定性指数 μ ∈ (0, 2) 解决一个（随机）目标问题。具体而言，给定一个概率密度函数，确定半群势，进而确定该概率密度函数实际上是主方程的长时间渐近（目标）解的跳跃型动力学。在此，不同 μ 运动情形的渐近行为不再依赖于 μ。通过考虑高斯和柯西族目标概率密度函数对此进行了举例说明。分析了反向工程的一个互补问题：先验给定一个半群势，量化相关主方程解的渐近行为以及不变概率密度函数本身对 μ 驱动选择的敏感程度。针对所谓的 Lévy 振荡器的 μ 族完成了此任务。
