  Using simple kinematical arguments, we derive the Fokker-Planck equation for
diffusion processes in curved spacetimes. In the case of Brownian motion, it
coincides with Eckart's relativistic heat equation (albeit in a simpler form),
and therefore provides a microscopic justification for his phenomenological
heat-flux ansatz. Furthermore, we obtain the small-time asymptotic expansion of
the mean square displacement of Brownian motion in static spacetimes. Beyond
general relativity itself, this result has potential applications in analogue
gravitational systems.
