Structural and evolutionary aspects of stochastic processes

Stochastic processes constitute a broad class of objects of central importance in complex systems. The main structural aspect of stochastic processes that is treated in this thesis is their hierarchical dynamics. The dynamics of a process is hierarchical if there is a projection from the process' state space to a smaller state space such that the projection imposes dynamics on the coarser level that is Markovian (i.e. that the probability distribution of future states is dependent only on the current state). A stochastic process may have several such projections that together form a hierarchy. Paper I describes an algorithm that automatically infers these hierarchical levels. The method operates on observed time series of states, and hence knowledge of the intrinsic workings of a process is not required (and typically not known a priori). The basic idea behind the method is to systematically explore the set of possible projections and to statistically determine which of these that imposes Markovian dynamics.</p> Paper II and III treat a conceptual model of pre-biotic evolution; a soup of stochastic processes that generate new stochastic processes. Despite the model's simplicity, it manages to demonstrate rich self-organized behavior. The soup is found to successively build higher levels of organization by autocatalysis, which is facilitated by local general-purpose components with low complexity. This agrees with the generally argued notion that hierarchical organization is a prerequisite for a system to evolve complexity. Another aspect of pre-biotic evolution is treated in Paper IV. It presents a preliminary study of the conditions under which groups of replicating molecules can form aggregated units of selection. The question is whether molecules that promote the growth of an aggregate at the cost of their own replication rate can remain in the system despite their low local fitness. Due to mechanisms resembling group selection, this is shown to be possible under certain conditions