The Aldous diffusion: a stationary evolution of the Brownian CRT

Abstract

Motivated by a down-up Markov chain on cladograms, David Aldous conjectured in 1999 that there exists a "diffusion on continuum trees" whose mass partitions at any finite number of branch points evolve as Wright-Fisher diffusions with some negative mutation rates, until some branch point disappears. Building on previous work on interval-partition-valued processes, we construct this conjectured process via a consistent system of stationary evolutions of binary trees with k labeled leaves and edges decorated with interval partitions. The interval partitions are scaled Poisson-Dirichlet interval partitions whose interval lengths record subtree masses. They also possess a diversity property that captures certain distances in the continuum tree. Continuously evolving diversities give access to continuously evolving continuum tree distances. The pathwise construction allows us to study this "Aldous diffusion" in the Gromov-Hausdorff-Prokhorov space of rooted, weighted R-trees. We establish the simple Markov property and path-continuity. The Aldous diffusion is stationary with the distribution of the Brownian continuum random tree. While the Brownian CRT is a.s. binary, we show that there is a dense null set of exceptional times when the Aldous diffusion has a ternary branch point, including stopping times at which the strong Markov property fails. Our construction relates to the two-parameter Chinese restaurant process, branching processes, and stable L\'evy processes, among other connections. Wright-Fisher diffusions and the aforementioned processes of Poisson-Dirichlet interval partitions arise as interesting projections of the Aldous diffusion. Finally, one can embed Aldous's stationary down-up Markov chain on cladograms in the Aldous diffusion and hence address a related conjecture by David Aldous by establishing a scaling limit theorem.