Lévy flights, dynamical duality and fractional quantum mechanics

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 a suitable analytic continuation in time procedure. 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. Schrödinger 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 Lévy 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). PACS numbers: 02.50.Ey, 05.20.-y, 05.40.Jc 1 Brownian motion inspirations 1.1 Diffusion-type processes and dynamical semigroups The Langevin equation for a one-dimensional stochastic diffusion process in an external conservative force field F = −(∇V ): ẋ = F (x) + √ 2Db(t), where b(t) stands for the normalized white noise 〈b(t)〉 = 0, 〈b(t′)b(t)〉 = δ(t− t′), gives rise to the corresponding Fokker-Planck equation for the probability density ρ(x, t): ∂tρ = D∆ρ−∇(Fρ) . (1) By means of a standard substitution ρ(x, t) = Ψ(x, t) exp[−V (x)/2D], [1], we pass to a generalized diffusion equation for an auxiliary function Ψ(x, t): ∂tΨ = D∆Ψ− V(x)Ψ (2) ∗Presented at the 21 Marian Smoluchowski Symposium on Statistical Physics