  We study boundary crossing probability in the context of stochastic
inflation. We prove that for a generic multi-field inflationary potential, the
probability that the inflaton reaches infinitely far regions in the field space
is critically dependent on the number of fields, being nonzero for more than
two fields, and zero otherwise. We also provide several examples where the
boundary crossing probability can be calculated exactly, most notably, for a
particular landscape of a two-field model with a multi-well potential.
