  We study the Brownian motion of a field where there are boundaries in the
inflationary field space. Both the field and the boundary undergo Brownian
motions with the amplitudes of the noises determined by the Hubble expansion
rate of the corresponding dS spacetime. This setup mimics models of inflation
in which curvature perturbation is induced from inhomogeneities generated at
the surface of end of inflation. The cases of the drift dominated regime as
well as the diffusion dominated regime are studied in details. We calculate the
first hitting probabilities as well as the mean number of e-folds for the field
to hit either of the boundaries in the field space. The implications for models
of inflation are reviewed.
