  We revisit a simple toy model of two scalar fields in de Sitter space,
playing the roles of "system" and "environment" degrees of freedom, which
interact with each other. We show that there are secular divergences in
physically relevant observables which arise solely from the non-Markovian part
of the memory kernel, contrary to popular belief that secular growth typically
comes from local terms in evolution equations. Nevertheless, we show that these
terms can still be non-perturbatively resummed, using the time-convolutionless
master equation formalism, which improves upon previous approximations. At the
same time, there are other physical quantities in the same model that are
dominated by local terms in the memory kernel. Therefore, we conclude that, for
cosmological backgrounds, either the dissipation or the noise kernel can end up
being dominated by non-local terms depending on the nature of the
system-environment coupling.
