2 Velocity-space Diffusion Process

We derive explicit forms of Markovian transition probability densities for the velocity space and phase-space Brownian motion of a charged particle in a constant magnetic field. An old-fashioned problem of the Brownian motion of a charged particle in a constant magnetic field has originated from studies of the diffusion of plasma across a magnetic field [1], [2] and nowadays, together with a free Brownian motion example, stands for a textbook illustration of how transport and auto-correlation functions should be computed in generic situations governed by the Langevin equation cf. [3] but also [4], [5]. To our knowledge, except for the paper [2] no attempt was made in the literature to give a complete characterization of the pertinent stochastic process. However a striking peculiarity of Ref. [2] is that the Brownian motion in a magnetic field is there described in terms of operator-valued (matrix-valued functions) probability distributions that involve fractional powers of matrices. In consequence, we have no clean relationship with the standard formalism of Kramers-Smoluchowski equations, nor ways to stay in conformity with the standard wisdom about probabilistic procedures valid in case of the free Brownian motion (Ornstein-Uhlenbeck process), cf. [6], [7], [8]. Therefore, we address an issue of the Brownian motion of a charged particle in a magnetic field anew, to unravel its features of a fully-fledged stochastic diffusion process. The standard analysis of the Brownian motion of a free particle employs the Langevin equation d − → u dt = −β − → u + − → A (t) where − → u denotes the velocity of the particle and the influence of the surrounding medium on the motion (random acceleration) of the particle is modeled by means of two independent contributions. A systematic part −β − → u represents a dynamical friction. The remaining fluctuating 1