Continuum limits of tree-valued Markov chains and algebraic measure trees

In some approaches to the reconstruction of phylogenetic trees, Markov chain Monte Carlo methods are used. These in turn use simple base-chains on the set of (binary) trees of a given size $N$. It is at least of mathematical interest (but might also help to understand properties of such Markov chains when the trees are large) to consider limit processes as $N$ tends to infinity and the time is suitably sped up. Here, we have to decide in which state space we are working, i.e., what kind of objects we want to consider as "continuum trees" in the limit, and what we mean by "limit".

One by now almost-classical approach is to work in a space of metric measure spaces, but while it has proven successful in some situations, it seems difficult to prove convergence in others. Motivated by a particular Markov chain, the Aldous chain on cladograms, where existence of a limit process has been conjectured almost two decades ago, we introduce an alternative state space. We define the objects by a "tree structure" (formalized by a branch-point map) instead of a metric structure and call them algebraic measure trees. In this new state space, we are able to prove convergence of the Aldous chain to a limit process with continuous paths.

(joint work with Anita Winter and Leonid Mytnik)