Subtree prune and re-graft: a reversible real tree valued Markov process

February, 2005
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Steven N. Evans and Anita Winter

We use Dirichlet form methods to construct and analyze a reversible Markov process, the stationary distribution of which is the Brownian continuum random tree. This process is inspired by the subtree prune and re-graft (SPR) Markov chains that appear in phylogenetic analysis. A key technical ingredient in this work is the use of a novel Gromov--Hausdorff type distance to metrize the space whose elements are compact real trees equipped with a probability measure. Also, the investigation of the Dirichlet form hinges on a new path decomposition of the Brownian excursion.

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