Stochastic Ultra Slow Roll Inflation
  We study the ultra slow roll model in the context of stochastic inflation.
Using stochastic $\delta N$ formalism, we calculate the mean number of
$e$-folds, the power spectrum, the bispectrum and the stochastic corrections
into these observables. We reproduce correctly the known leading classical
contributions to these cosmological observables while we show that the
fractional corrections to cosmological observables induced from stochastic
dynamics are at the order of power spectrum. In addition, we consider a
hypothetical setup containing two absorbing barriers on both sides of the field
configuration and calculate the probability of first boundary crossing
associated with the classical motion and quantum jumps. This analysis includes
the limit of Brownian motion of the quantum fluctuations of a test scalar field
in a dS spacetime.
