  We construct a model of Brownian Motion on a pseudo-Riemannian manifold
associated with general relativity. There are two aspects of the problem: The
first is to define a sequence of stopping times associated with the Brownian
"kicks" or impulses. The second is to define the dynamics of the particle along
geodesics in between the Brownian kicks. When these two aspects are taken
together, we can associate various distributions with the motion. We will find
that the statistics of space-time events will obey a temperature dependent four
dimensional Gaussian distribution defined over the quaternions which locally
can be identified with Minkowski space. Analogously, the statistics of the
4-velocities will obey a kind of Maxwell-Juttner distribution. In contrast to
previous work, our processes are characterized by two independent proper time
variables defined with respect to the laboratory frame: a discrete one
corresponding to the stopping times when the impulses take place and a
continuous one corresponding to the geodesic motion in-between impulses. The
subsequent distributions are then solutions of partial differential equations
which contain derivatives with respect to both time variables.
