  We consider two processes that have been used to study gene duplication,
Watterson's [Genetics 105 (1983) 745--766] double recessive null model and
Lynch and Force's [Genetics 154 (2000) 459--473] subfunctionalization model.
Though the state spaces of these diffusions are two and six-dimensional,
respectively, we show in each case that the diffusion stays close to a curve.
Using ideas of Katzenberger [Ann. Probab. 19 (1991) 1587--1628] we show that
one-dimensional projections converge to diffusion processes, and we obtain
asymptotics for the time to loss of one gene copy. As a corollary we find that
the probability of subfunctionalization decreases exponentially fast as the
population size increases. This rigorously confirms a result Ward and Durrett
[Theor. Pop. Biol. 66 (2004) 93--100] found by simulation that the likelihood
of subfunctionalization for gene duplicates decays exponentially fast as the
population size increases.
