Spectator field dynamics in de Sitter and curvaton initial conditions

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.