# Relativistic diffusions

by Jacques Franchi (Univ. Strasbourg)

The classical theory of Brownian motion is not compatible with relativity, the heat flow propagating instantly till infinity. Around forty years ago, Dudley defined, on Minkowski space-time of special relativity, a substitute for the Laplace operator, which satisfies the Lorentz invariance, and showed its unicity.

The associated diffusion process undergoes continuously infinitesimal Lorentzian boosts, in random directions. Its velocity is a hyperbolic Brownian motion. Dudley showed that his diffusion selects a random asymptotic direction, asymptotically with the velocity of light. Recently, Yves Le Jan and myself proposed a natural extension of the relativistic diffusion to general relativity, that is, to the framework of Lorentzian manifolds. This is analogue to the extension of Euclidian Brownian motion to Riemannian manifolds. This general diffusion can be constructed either by means of the frame bundle, or by developing the flat Dudley diffusion. Its infinitesimal generator is the perturbation of the geodesic flow vector field by means of the vertical Laplacian of the manifold.

It has already been studied in Schwarzschild and Gödel universes.