Loewner chains and evolution families on parallel slit half-planes

In this paper, we define and study Loewner chains and evolution families on finitely multiply-connected domains in the complex plane. These chains and families consist of conformal mappings on parallel slit half-planes and have one and two “time” parameters, respectively. By analogy with the case of simply connected domains, we develop a general theory of Loewner chains and evolution families on multiply connected domains and, in particular, prove that they obey the chordal Komatu–Loewner differential equations driven by measure-valued processes. Our method involves Brownian motion with darning, as do some recent studies.