Here is my Part III essay. It requires some knowledge of geometry and more so in probability. Please post the mistakes here.
The paper is about random motions in spacetime (mainly the Minkowski space) and is divided into three major parts. In the first part the notion of Levy processes on Lie groups is developed with the aid of stochastic calculus and semi-groups of operators. I think this part could be safely skipped.
The second part involves constructing a Brownian motion on Lorentz and Minkowski spaces. The idea is to use the orthonormal frame bundle to construct the diffusion and project it down the the manifold.
The last part is about the specifics of the asymptotic behaviour of a relativistic diffusion in a Minkowski space. Some interesting result include that the angular part of the diffusion converges but the radial part is transient. Also the diffusion approaches a hyperplane which also implies the existence of non-constant bounded harmonic functions.
It may be a little heavy going but I hope it will be a good read for somebody here. Feedback and suggestions is always welcome.