  Processes of random tessellations of the Euclidean space $\mathbb{R}^d$,
$d\geq 1$, are considered which are generated by subsequent division of their
cells. Such processes are characterized by the laws of the life times of the
cells until their division and by the laws for the random hyperplanes that
divide the cells at the end of their life times. The STIT tessellation
processes are a reference model. In the present paper a generalization
concerning the life time distributions is introduced, a sufficient condition
for the existence of such cell division tessellation processes is provided and
a construction is described. In particular, for the case that the random
dividing hyperplanes have a Mondrian distribution -- which means that all cells
of the tessellations are cuboids -- it is shown that the intrinsic volumes,
except the Euler characteristic, can be used as the parameter for the
exponential life time distribution of the cells.
