  This paper contains a detailed study of the properties of a simple model
attempting to explain dark energy as originated from quantum fluctuations of a
light spectator scalar field in inflation. In [1] we recently outlined how
Starobinsky's stochastic formalism can be used to study the spatial
correlations imprinted on dark energy by its quantum origin in this model and
we studied their possible role in relieving the Hubble tension. Here we provide
a more comprehensive derivation of the results in [1] and we refine some of our
estimates, comparing to the approximate results obtained previously. Among the
main results, we analyze the non-coincident correlators predicted by a full
field theoretical treatment and their relation with those computed within the
stochastic formalism. We find that in the region where stochastic theory
predicts significant sub-Hubble correlators it is in disagreement with field
theoretical predictions. However, agreement can be restored by introducing a
reduced speed of sound for the scalar field. We also discuss an alternative
approach to the problem of studying correlators within the stochastic formalism
based directly on the evolution of probability distributions. We find that the
two approaches give the same answer for 2-point functions of the field, but not
for 4-point functions relevant to density correlators and we discuss the
behaviour of the two methods with respect to Wick's theorem.
