  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).
