  We use the spectral representation of the stochastic Starobinsky-Yokoyama
approach to compute correlation functions in de Sitter space for a scalar field
with a symmetric or asymmetric double-well potential. The terms in the spectral
expansion are determined by the eigenvalues and eigenfunctions of the
time-independent Fokker-Planck differential operator, and we solve them
numerically. The long-distance asymptotic behaviour is given by the lowest
state in the spectrum, but we demonstrate that the magnitude of the coeffients
of different terms can be very different, and the correlator can be dominated
by different terms at different distances. This can give rise to potentially
observable cosmological signatures. In many cases the dominant states in the
expansion do not correspond to small fluctuations around a minimum of the
potential and are therefore not visible in perturbation theory. We discuss the
physical interpretation these states, which can be present even when the
potential has only one minimum.
