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The value of a continuous character evolving on a phylogenetic tree iscommonly modelled as the location of a particle moving under one-dimensionalBrownian motion with constant rate. The Brownian motion model is best suited tocharacters evolving under neutral drift or tracking an optimum that driftsneutrally. We present a generalization of the Brownian motion model whichrelaxes assumptions of neutrality and gradualism by considering increments toevolving characters to be drawn from a heavy-tailed stable distribution (ofwhich the normal distribution is a specialized form). We describe Markov chainMonte Carlo methods for fitting the model to biological data paying specialattention to ancestral state reconstruction, and study the performance of themodel in comparison with a selection of existing comparative methods, usingboth simulated data and a database of body mass in 1,679 mammalian species. Wediscuss hypothesis testing and model selection. The new model is well suited toa stochastic process with a volatile rate of change in which biologicalcharacters undergo a mixture of neutral drift and occasional evolutionaryevents of large magnitude.

Inferring ancestral states without assuming neutrality or gradualism using a stable model of continuous character evolution