  Scalar fields interacting with the primordial curvature perturbation during
inflation may communicate their statistics to the latter. This situation
motivates the study of how the probability density function (PDF) of a light
spectator field $\varphi$ in a pure de Sitter space-time, becomes non-Gaussian
under the influence of a scalar potential ${\mathcal V(\varphi)}$. One approach
to this problem is offered by the stochastic formalism introduced by
Starobinsky and Yokoyama. It results in a Fokker-Planck equation for the
time-dependent PDF $\rho (\varphi , t)$ describing the statistics of $\varphi$
which, in the limit of equilibrium gives one back the solution $\rho (\varphi)
\propto \exp \big[ - \frac{8 \pi^2}{3 H^4} {\mathcal V(\varphi)} \big]$. We
study the derivation of $\rho (\varphi , t)$ using quantum field theory tools.
Our approach yields an almost Gaussian distribution function, distorted by
minor corrections comprised of terms proportional to powers of $\Delta N \times
\mathcal O(\partial_\varphi) {\mathcal V(\varphi)}$, where $\Delta N$ is the
number of $e$-folds succeeding the Hubble-horizon crossing of $\varphi$'s
wavelengths, and $\mathcal O(\partial_\varphi)$ stands for a derivative
operator acting on ${\mathcal V(\varphi)}$. This general form is obtained
perturbatively and remains valid even with loop corrections. Our solution
satisfies a Fokker-Planck equation that receives corrections with respect to
the one found within the stochastic approach, allowing us to comment on the
validity of the standard equilibrium solution for generic potentials. We posit
that higher order corrections to the Fokker-Planck equation may become
important towards the equilibrium.
