Continuous branching structures as scaling limits of discrete branching structures

We consider branching structures, a particular class of probabilistic models used to represent an evolving population, in which each individual produces a random number of offspring, according to a fixed probability distribution and independently of the other individuals. In particular, we consider four types of branching structure: discrete/ continuous branching processes and discrete/continuous branching genealogical structures (where with discrete/continuous we intend structures with discrete/continuous-time and discrete/continuous-state space). The goal of the thesis is to present some of the results found in literature and studied throughout years by many authors, re-elaborating them in a single work, with the aim of highlighting the relations between the different branching structures. In particular, we focus on the relations between discrete and continuous branching structures, explaining under which conditions continuous branching structures can be obtained as the continuous approximations of discrete ones, or more formally, showing under which conditions continuous branching structures can be obtained as scaling limits of discrete ones (that is: when a sequence of discrete branching structures, once rescaled in space and time in some sense, somehow converges to a continuous branching structure). In studying these relations, we focus on non-supercritical branching structures, a subclass of branching structures used to represent populations with finite lifetime. This restriction allows us to give a uniform description of the relations between the different types of branching structure we introduce. Throughout the work, we also provide many explanatory examples, in order to help the reader to understand the developed objects, concepts and results in depth.