  The Aldous diffusion is a conjectured Markov process on the space of real
trees that is the continuum analogue of discrete Markov chains on binary trees.
We construct this conjectured process via a consistent system of stationary
evolutions of binary trees with $k$ labeled leaves and edges decorated with
diffusions on a space of interval partitions constructed in previous work by
the same authors. This pathwise construction allows us to study and compute
path properties of the Aldous diffusion including evolutions of projected
masses and distances between branch points. A key part of proving the
consistency of the projective system is Rogers and Pitman's notion of
intertwining.
