A covariant Fokker-Planck-type equation for a simple gas and an equation for the Brownian motion are derived from a relativistic kinetic theory based on the Boltzmann equation. For the simple gas the dynamic friction four-vector and the diffusion tensor are identified and written in terms of integrals which take into account the collision processes. In the case of Brownian motion, the Brownian particles are considered as nonrelativistic, whereas the background gas behaves as a relativistic gas. A general expression for the semirelativistic viscous friction coefficient is obtained and the particular case of constant differential cross section is analyzed for which the nonrelativistic and ultrarelativistic limiting cases are calculated.