We study a family of n-dimensional diffusions, taking values in the unit
simplex of vectors with nonnegative coordinates that add up to one. These
processes satisfy stochastic differential equations which are similar to the
ones for the classical Wright-Fisher diffusions, except that the "mutation
rates" are now nonpositive. This model, suggested by Aldous, appears in the
study of a conjectured diffusion limit for a Markov chain on Cladograms. The
striking feature of these models is that the boundary is not reflecting, and we
kill the process once it hits the boundary. We derive the explicit exit
distribution from the simplex and probabilistic bounds on the exit time. We
also prove that these processes can be viewed as a "stochastic time-reversal"
of a Wright-Fisher process of increasing dimensions and conditioned at a random
time. A key idea in our proofs is a skew-product construction using certain
one-dimensional diffusions called Bessel-square processes of negative
dimensions, which have been recently introduced by Going-Jaeschke and Yor.Comment: Published in at http://dx.doi.org/10.1214/11-AOP704 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
