Topological spaces of trees as state spaces for stochastic processes

This thesis presents some aspects of the theory of continuum limits of tree-valued stochastic processes, as well as limits of stochastic processes on (possibly random) trees as the size of the underlying trees tends to infinity. In both situations, in order to get the full picture, one has to work in a suitable, sufficiently nice, abstract topological state space of (continuum) trees. The focus of the articles collected in this thesis is on the construction and analysis of such state spaces, as well as their usage to obtain general limit theorems.

The state spaces fall into two categories. Firstly, the more classical framework using metric measure spaces and various generalizations. Secondly, a new, more algebraic/combinatorial approach of coding the tree-structure by a branch point map without reference to a canonical or pre-specified metric. We call these objects algebraic measure trees. An important classical tool is to code metric measure trees by excursions. Similarly, we can code binary algebraic measure trees by triangulations of the circle.

Some stochastic processes are considered as examples to highlight the applicability and usefulness of the developed state spaces and tools. These are tree-valued multi-type Fleming-Viot processes, tree-valued pruning processes, random walks on trees, and the Aldous chain on cladograms. In all these cases we obtain limit theorems in suitable state spaces of measure trees.


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