Smooth coarse-graining and colored noise dynamics in stochastic
  inflation
  We consider stochastic inflation coarse-grained using a general class of
exponential filters. Such a coarse-graining prescription gives rise to
inflaton-Langevin equations sourced by colored noise that is correlated in
$e$-fold time. The dynamics are studied first in slow-roll for simple
potentials using first-order perturbative, semi-analytical calculations which
are later compared to numerical simulations. Subsequent calculations are
performed using an exponentially correlated noise which appears as a leading
order correction to the full slow-roll noise correlation functions of the type
$\big\langle \xi(N)\xi(N') \big\rangle_{(n)}\sim\left( \cosh\left[ n(N-N')+1
\right] \right)^{-1}$. We find that the power spectrum of curvature
perturbations $\mathcal{P}_{\zeta}$ is suppressed at early $e$-folds, with the
suppression controlled by $n$. Furthermore, we use the leading order,
exponentially correlated noise and perform a first passage time analysis to
compute the statistics of the stochastic $e$-fold distribution $\mathcal{N}$
and derive an approximate expression for the mean number of $e$-folds
$\big\langle \mathcal{N} \big\rangle$. Comparing analytical results with
numerical simulations of the inflaton dynamics, we show that the leading order
noise correlation function can be used as a very good approximation of the
exact noise, the latter being more difficult to simulate.
