  A self-consistent and universal description of friction and diffusion for
Brownian particles (grains) in different systems, as a gas with Boltzmann
collisions, dusty plasma with ion absorption by grains, and for active
particles (e.g., cells in biological systems) is suggested on the basis of the
appropriate Fokker-Planck equation. Restrictions for application of the
Fokker-Planck equation to the problem of velocity-dependent friction and
diffusion coefficients are found. General description for this coefficient is
formulated on the basis of master equation. Relation of the diffusion
coefficient in the coordinate and velocity spaces is found for active (capable
to transfer momentum to the ambient media) and passive particles in the
framework of the Fokker-Planck equation.
  The problem of anomalous space diffusion is formulated on the basis of the
appropriate probability transition (PT) function. The method of partial
differentiation is avoided to construct the correct probability distributions
for arbitrary distances, what is important for applications to different
stochastic problems. Generale equation for time-dependent PT function is
formulated and discussed.
  Generalized friction in the velocity space is determined and applied to
describe the friction force itself as well as the drag force in the case of a
non-zero driven ion velocity in plasmas. The negative friction due to ion
scattering on grains exists and can be realized for the appropriate
experimental conditions.
