  We discuss general positivity conditions necessary for a definition of a
relativistic diffusion on the phase space. We show that Lorentz covariant
random vector fields on the forward cone $p^{2}\geq 0$ lead to a definition of
a generator of Lorentz covariant diffusions. We discuss in more detail
diffusions arising from particle dynamics in a random electromagnetic field
approximating the quantum field at finite temperature. We develop statistical
mechanics of a gas of diffusing particles. We discuss viscosity of such a gas
in an expansion in gradients of the fluid velocity.
