The Aldous chain on cladograms in the diffusion limit

TL;DR

This paper proves the convergence of a Markov chain on cladograms to a continuous Aldous diffusion process, characterizes its properties, and provides explicit results on subtree mass distributions.

Contribution

It rigorously constructs the Aldous diffusion as a limit of Markov chains on cladograms and analyzes its properties and subtree mass dynamics.

Findings

Aldous diffusion is a Feller process with continuous paths.

The algebraic measure Brownian CRT is the unique invariant distribution.

Explicit infinitesimal evolution of subtree masses under the diffusion.

Abstract

In [Ald00], Aldous investigates a symmetric Markov chain on cladograms and gives bounds on its mixing and relaxation times. The latter bound was sharpened in [Sch02]. In the present paper we encode cladograms as binary, algebraic measure trees and show that this Markov chain on cladograms with fixed number of leaves converges in distribution as the number of leaves goes to infinity. We give a rigorous construction of the limit, whose existence was conjectured by Aldous and which we therefore refer to as Aldous diffusion, as a solution of a well-posed martingale problem. We show that the Aldous diffusion is a Feller process with continuous paths, and the algebraic measure Brownian CRT is its unique invariant distribution. Furthermore, we consider the vector of the masses of the three subtrees connected to a sampled branch point. In the Brownian CRT, its annealed law is known to be the Dirichlet distribution. Here, we give an explicit expression for the infinitesimal evolution of its quenched law under the Aldous diffusion.