  Stochastic effects in multi-field inflationary scenarios are investigated. A
hierarchy of diffusion equations is derived, the solutions of which yield
moments of the numbers of inflationary $e$-folds. Solving the resulting partial
differential equations in multi-dimensional field space is more challenging
than the single-field case. A few tractable examples are discussed, which show
that the number of fields is, in general, a critical parameter. When more than
two fields are present for instance, the probability to explore arbitrarily
large-field regions of the potential, otherwise inaccessible to single-field
dynamics, becomes non-zero. In some configurations, this gives rise to an
infinite mean number of $e$-folds, regardless of the initial conditions.
Another difference with respect to single-field scenarios is that multi-field
stochastic effects can be large even at sub-Planckian energy. This opens
interesting new possibilities for probing quantum effects in inflationary
dynamics, since the moments of the numbers of $e$-folds can be used to
calculate the distribution of primordial density perturbations in the
stochastic-$\delta N$ formalism.
