  K. G\"odel [G1] discovered his celebrated solution to Einstein equations in
1949. Additional contributions were made by Kundt [K] and Hawking-Ellis
([H-E],5.7). On the other hand, a general Lorentz invariant operator,
associated to the so-called "relativistic diffusion'', and making sense in any
Lorentz manifold, was introduced by Franchi-Le Jan in [F-LJ]. Here is purposed
a first study of the relativistic diffusion in the framework of G\"odel's
universe, which contains matter.
