Abstract

Relativistic diffusions are models of random motion in spacetime of an object moving with a speed less than the speed of light. These processes are the Lorentzian analogues of Brownian motion in a Riemannian context. In so far as they are defined in purely geometric terms, it is very likely that part (or all?) of the geometry of the ambient spacetime may be recovered from the probablistic behaviour of these processes. In a Riemannian setting, this probabilistic view on geometry is well-illustrated by Weyl and Pleyel formulas for the heat kernel of Brownian motion where local and global informations about the geometry appear.

We shall investigate in this talk one aspect of this geometry/probability correspondence. Dating back to Penrose and Hawking’s results, it is now well-established that the appearance of singularities in Einstein’s theory of gravitation is unavoidable under quite natural assumptions. Although there is no definitive agreement on what should be called a singularity of spacetime, a largely used notion of singularity is the existence in spacetime of incomplete geodesics. Is there a link between geodesic and probabilistic incompleteness? This will be the main question will shall adress.

http://www.statslab.cam.ac.uk/~ismael/