Non-Markovian harmonic oscillator across a magnetic field and time-dependent force fields

We study the non-Markovian Brownian motion of an electrically charged harmonic oscillator through the action of both a constant magnetic field and time-dependent force fields. The generalized Langevin equation with a friction memory kernel is used to derive the generalized phase-space Fokker-Planck equation for the harmonic oscillator in the absence and in the presence of time-dependent force fields. To achieve our goal, the characteristic function method is applied to obtain, in an accurate way, the theoretical description of the problem. We explicitly calculate the correlation and cross-correlation functions for the position and velocity vectors. We show that the relevant physics behind the theory is contained in the generalized diffusion coefficient, which accounts for the natural coupling between both the harmonic oscillator and magnetic field. Our theoretical results are compared with those previously reported in the literature.