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The Aldous diffusion is a conjectured Markov process on the space of realtrees that is the continuum analogue of discrete Markov chains on binary trees.We construct this conjectured process via a consistent system of stationaryevolutions of binary trees with $k$ labeled leaves and edges decorated withdiffusions on a space of interval partitions constructed in previous work bythe same authors. This pathwise construction allows us to study and computepath properties of the Aldous diffusion including evolutions of projectedmasses and distances between branch points. A key part of proving theconsistency of the projective system is Rogers and Pitman's notion ofintertwining.

Aldous diffusion I: a projective system of continuum $k$-tree evolutions