  Strongly continuous semigroups of unital completely positive maps (i.e.
quantum Markov semigroups or quantum dynamical semigroups) on compact quantum
groups are studied. We show that quantum Markov semigroups on the universal or
reduced C${}^*$-algebra of a compact quantum group which are translation
invariant (w.r.t. to the coproduct) are in one-to-one correspondence with
L\'evy processes on its $*$-Hopf algebra. We use the theory of L\'evy processes
on involutive bialgebras to characterize symmetry properties of the associated
quantum Markov semigroup. It turns out that the quantum Markov semigroup is
GNS-symmetric (resp. KMS-symmetric) if and only if the generating functional of
the L\'evy process is invariant under the antipode (resp. the unitary
antipode). Furthermore, we study L\'evy processes whose marginal states are
invariant under the adjoint action. In particular, we give a complete
description of generating functionals on the free orthogonal quantum group
$O_n^+$ that are invariant under the adjoint action. Finally, some aspects of
the potential theory are investigated. We describe how the Dirichlet form and a
derivation can be recovered from a quantum Markov semigroup and its L\'evy
process and we show how, under the assumption of GNS-symmetry and using the
associated Sch\"urmann triple, this gives rise to spectral triples. We discuss
in details how the above results apply to compact groups, group C$^*$-algebras
of countable discrete groups, free orthogonal quantum groups $O_n^+$ and the
twisted $SU_q (2)$ quantum group.
