  We investigate the stochastic behaviour of long wavelength modes of light
spectator scalar fields during inflation. When starting from a classical field
value, the probability distribution for the spectator both spreads out and
moves towards an equilibrium distribution. We study the timescales for a mixed
quadratic and quartic potential. The timescale of equilibration depends on the
parameters of the model, and can be surprisingly large, even much more than
thousands of e-folds. These results imply that the initial conditions for
spectator fields are not automatically erased during inflation. Applying the
results to the curvaton model, we calculate the probability distribution of the
curvature perturbation and discuss 'typical' Universes.
