  We calculate the curvature power spectrum sourced by spectator fields that
are excited repeatedly and non-adiabatically during inflation. In the absence
of detailed information of the nature of spectator field interactions, we
consider an ensemble of models with intervals between the repeated interactions
and interaction strengths drawn from simple probabilistic distributions. We
show that the curvature power spectra of each member of the ensemble shows rich
structure with many features, and there is a large variability between
different realizations of the same ensemble. Such features can be probed by the
cosmic microwave background (CMB) and large scale structure observations. They
can also have implications for primordial black hole formation and CMB spectral
distortions.
  The geometric random walk behavior of the spectator field allows us to
calculate the ensemble-averaged power spectrum of curvature perturbations
semi-analytically. For sufficiently large stochastic sourcing, the
ensemble-averaged power spectrum shows a scale dependence arising from the time
spent by modes outside the horizon during the period of particle production, in
spite of there being no preferred scale in the underlying model. We find that
the magnitude of the ensemble-averaged power spectrum overestimates the typical
power spectra in the ensemble because the ensemble distribution of the power
spectra is highly non-Gaussian with fat tails.
