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We study the boundary-crossing probability in the context of stochastic inflation. We prove that for a generic multifield 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 in which the boundary-crossing probability can be calculated exactly, most notably for a particular landscape of a two-field model with a multiwell potential
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