  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.
