Local duality in loewner equations

Among diversity of frameworks and constructions introduced in Loewner theory by different authors, one can distinguish two closely related but still different ways of reasoning, which colloquially may be described as "increasing" and "decreasing". In this paper we review in short the main types of (deterministic) Loewner evolution discussed in the literature and describe in detail the local duality between "increasing" and "decreasing" cases within the general unifying approach in Loewner theory proposed recently in [7, 8, 9]. In particular, we extend several results of [6], which deals with the chordal Loewner evolution, to this general setting. Although the duality is given by a simple change of the parameter, not all the results for the "decreasing" case can be obtained by mere translating the corresponding results for the "increasing" case. In particular, as a byproduct of establishing local duality between evolution families and their "decreasing" counterparts we obtain a new characterization of generalized Loewner chains.