  This paper defines the notion of generators for a class of decreasing radial
Loewner chains which are only continuous with respect to time. For this
purpose, "Loewner's integral equation" which generalizes Loewner's differential
equation is defined and analyzed. The definition of generators is motivated by
the L\'evy-Khintchine representation for additive processes on the unit circle.
Actually, we can and do introduce a homeomorphism between the above class of
Loewner chains and the set of the distributions of increments of additive
processes equipped with suitable topologies. On the other hand, from the
viewpoint of non-commutative probability theory, the above generators also
induce bijections with some other objects: in particular, monotone convolution
hemigroups and free convolution hemigroups. Finally, the generators of Loewner
chains constructed from free convolution hemigroups via subordination are
computed.
