Levy targeting and the principle of detailed balance

TL;DR

This paper explores the dynamics of confined Lévy flights under detailed balance, introducing a method to design jump processes targeting specific invariant distributions for arbitrary stability indices.

Contribution

It develops a novel ta-targeting approach for Lévy flights and solves the reverse engineering problem for Lévy oscillators with quadratic potentials.

Findings

Successfully target invariant PDFs for Lévy flights with various stability indices

Provides explicit solutions for Lévy oscillators with quadratic potentials

Establishes a connection between Lévy processes and Schrf6dinger semigroup dynamics.

Abstract

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 $\mu$.