We derive explicit forms of Markovian transition probability densities for the velocity space, phase-space and the Smoluchowski configuration-space Brownian motion of a charged particle in a constant magnetic field. By invoking a hydrodynamical formalism for those stochastic processes, we quantify a continual (net on the local average) heat transfer from the thermostat to diffusing particles.

We investigate an undamped random phase-space dynamics in deterministic external force fields (conservative and magnetic ones). By employing the hydrodynamical formalism for those stochastic processes we analyze microscopic kinetic-type "collision invariants" and their relationship to local conservation laws (moment equations) in the fully nonequlibrium context. We address an issue of the continual heat absorption (particles "energization") in the course of the process and its possible physical implementations. IntroductionThe major topic discussed in the present paper will be the frictionless random motion which is introduced in close affinity with second-order processes driven by white noise, [1,2]. In the course of motion an unrestricted "energization" of particles is possible, a phenomenon which has not received much attention in the literature, although appears to be set on convincing phenomenological grounds in specific (nontypical) physical surroundings, [3,4].A classic problem of an irreversible behaviour of a particle embedded in the large encompassing system, [5,6,7] pertains to the random evolution of tracer particles in a fluid (like e.g. those performing Brownian motion). In that case, a particle -bath coupling is expected to imply quick relaxation towards a stationary probability distribution (equilibrium Maxwellian for a given a priori temperature T of the surrounding fluid/bath) and thus making the system amenable to standard fluctuationdissipation theorems. The dissipation of randomly acquired energy back to the bath is here enforced by dynamical friction mechanisms, [8,9,10,7], although no balance between stochastic forcing (random *

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