  It is becoming increasingly clear that large but rare fluctuations of the
primordial curvature field, controlled by the tail of its probability
distribution, could have dramatic effects on the current structure of the
universe -- {\it e.g.} via primordial black-holes. However, the use of standard
perturbation theory to study the evolution of fluctuations during inflation
fails in providing a reliable description of how non-linear interactions induce
non-Gaussian tails. Here, we use the stochastic inflation formalism to study
the non-perturbative effects from multi-field fluctuations on the statistical
properties of the primordial curvature field. Starting from the effective
action describing multi-field fluctuations, we compute the joint probability
density function and show that enhanced non-Gaussian tails are a generic
feature of slow-roll inflation with additional degrees of freedom.
