  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.
