Cauchy flights in confining potentials

TL;DR

This paper explores methods to design stochastic processes with specific stationary distributions for Levy flights in confining potentials, comparing Langevin and Le9vy-Schrf6dinger approaches, exemplified by Cauchy processes.

Contribution

It introduces a framework for reverse engineering Levy flight processes with desired stationary densities using two distinct stochastic modeling approaches.

Findings

Demonstrates the design of Levy flight processes with targeted stationary distributions.

Compares Langevin and Le9vy-Schrf6dinger dynamics for confining Levy flights.

Uses Cauchy processes for illustrative computational examples.

Abstract

We analyze confining mechanisms for L\'evy flights evolving under an influence of external potentials. Given a stationary probability density function (pdf), we address the reverse engineering problem: design a jump-type stochastic process whose target pdf (eventually asymptotic) equals the preselected one. To this end, dynamically distinct jump-type processes can be employed. We demonstrate that one "targeted stochasticity" scenario involves Langevin systems with a symmetric stable noise. Another derives from the L\'evy-Schr\"odinger semigroup dynamics (closely linked with topologically induced super-diffusions), which has no standard Langevin representation. For computational and visualization purposes, the Cauchy driver is employed to exemplify our considerations.