  L\'evy processes in the sense of Sch\"urmann on the Lie algebra of the
Lorentz grouop are studied. It is known that only one of the irreducible
unitary representations of the Lorentz group admits a non-trivial one-cocycle.
A Sch\"urmann triple is constructed for this cocycle and the properties of the
associated L\'evy process are investigated. The decommpositions of the
restrictions of this triple to the Lie subalgebras $so(3)$ and $so(2,1)$ are
described.
