where each ( ) ∈ , 1 ≤  ≤ , and  is arbitrary. For convenience, define (0 0) = (0 0) and (+1 +1) = (1 1). Define such a point sequence to be an up/right path if, for any  ≥ 1, we have −1 ≤  and −1 ≤ . Hence an up/right path joins points of  in a continuous, piecewise linear manner with line segments of slope , 0 ≤  ≤ ∞, attaching (−1 −1) and ( ) for al...

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 fr...