On the Aldous diffusion on Continuum Trees. I

Abstract

Consider a Markov chain on the space of rooted real binary trees that randomly removes leaves and reinserts them on a random edge and suitably rescales the lengths of edges. This chain was introduced by David Aldous who conjectured a diffusion limit of this chain, as the size of the tree grows, on the space of continuum trees. We prove the existence of a process on continuum trees, which via a random time change, displays properties one would expect from the conjectured Aldous diffusion. The existence of our process is proved by considering an explicit scaled limit of a Poissonized version of the Aldous Markov chain running on finite trees. The analysis involves taking limit of a sequence of splitting trees whose age processes converge but the contour process does not. Several formulas about the limiting process are derived.