  Monotone L\'evy processes with additive increments are defined and studied.
It is shown that these processes have a natural Markov structure and their
Markov transition semigroups are characterized using the monotone
L\'evy-Khintchine formula. Monotone L\'evy processes turn out to be related to
classical L\'evy processes via Attal's ``remarkable transformation.'' A
monotone analogue of the family of exponential martingales associated to a
classical L\'evy process is also defined.
