  The value of a continuous character evolving on a phylogenetic tree is
commonly modelled as the location of a particle moving under one-dimensional
Brownian motion with constant rate. The Brownian motion model is best suited to
characters evolving under neutral drift or tracking an optimum that drifts
neutrally. We present a generalization of the Brownian motion model which
relaxes assumptions of neutrality and gradualism by considering increments to
evolving characters to be drawn from a heavy-tailed stable distribution (of
which the normal distribution is a specialized form). We describe Markov chain
Monte Carlo methods for fitting the model to biological data paying special
attention to ancestral state reconstruction, and study the performance of the
model in comparison with a selection of existing comparative methods, using
both simulated data and a database of body mass in 1,679 mammalian species. We
discuss hypothesis testing and model selection. The new model is well suited to
a stochastic process with a volatile rate of change in which biological
characters undergo a mixture of neutral drift and occasional evolutionary
events of large magnitude.
