Levy flights, dynamical duality and fractional quantum mechanics

TL;DR

This paper explores a duality in diffusion processes, including Levy flights, by analytically continuing the evolution in time, linking non-Hermitian operators to Hermitian ones, and extending the concept to fractional quantum mechanics.

Contribution

It introduces a novel duality framework connecting classical diffusion and quantum-like fractional dynamics through analytic continuation in time.

Findings

Established a duality between diffusion and quantum dynamics.

Extended duality to Levy flights with external forcing.

Demonstrated the applicability of fractional dynamical semigroups.

Abstract

We discuss dual time evolution scenarios which, albeit running according to the same real time clock, in each considered case may be mapped among each other by means of an analytic continuation in time. This dynamical duality is a generic feature of diffusion-type processes. Technically that involves a familiar transformation from a non-Hermitian Fokker-Planck operator to the Hermitian operator (e.g. Schroedinger Hamiltonian), whose negative is known to generate a dynamical semigroup. Under suitable restrictions upon the generator, the semigroup admits an analytic continuation in time and ultimately yields dual motions. We analyze an extension of the duality concept to Levy flights, free and with an external forcing, while presuming that the corresponding evolution rule (fractional dynamical semigroup) is a dual counterpart of the quantum motion (fractional unitary dynamics).