Metabasins or a State Space Aggregation for Finite Markov Chains with exponentially small Transition Probabilities

The present work is devoted to a phenomenon known from physics and chemistry that is hitherto studied only by means of computer simulations and neither there nor in mathematics rigorously defined: the notion of metabasins. Metabasins are a partition of the state space of certain physical systems according to specific aggregation rules along a given finite simulation path. The main challenge of this thesis lies in the construction and analysis of a path-independent approach providing certain metabasin-intrinsic properties (see Properties 1–5 in the Introduction). These are for instance the occurrence of specific sojourn times, the absence of multiple forward-backward jumps, the independence of entrance and exit state, or the similarity of energy barriers between different basins. The study of this problem is done within the framework of ergodic, reversible finite Markov chains with exponentially small transition probabilities depending on some energy function. In the first part of this thesis, a definition of metabasins is developed, which relies on the well established theory of metastability and complies with Properties 1–5. These metabasins basically emerge as valleys and unions of valleys of the energy landscape. Unlike similar works on this topic, valleys of completely different order are considered. Having introduced and analyzed those valleys and the notion of stability which is immanent to them in detail, the requested properties are derived. For this purpose, the process behavior on single valleys and the transitions between them are entirely specified. More specifically, typical trajectories on single valleys are determined, the average sojourn times are identified to depend on the depth of the valley in an exponential manner, and a certain aggregated process is defined that detects only the current valley and neglects the specific state therein. For this process, an asymptotic (semi-)Markov property is proved and its transition probabilities are determined. Using these probabilities, multiple forward-backward jumps are shown to be quite unlikely if and only if the energy barriers are approximately of the same height. The second part of this work is addressed to the goodness of the aggregation and studies the accordance with the path-dependent definition as well as the impact of the aggregation on the mixing-, cover-, and hitting times. The probability of accordance of both definitions is described by means of certain parameters measuring the degree of disorder in the system. For highly disordered systems arises a high probability of accordance. Furthermore, it is proved that the impact of the …