  We show that if either the process is strong Feller and the boundary point is
probabilistically regular for the stopping set, or the process is strong Markov
and the boundary point is probabilistically regular for the interior of the
stopping set, then the boundary point is Green regular for the stopping set.
Combining this implication with the existence of a continuously differentiable
flow of the process we show that the value function is continuously
differentiable at the optimal stopping boundary whenever the gain function is
so. The derived fact holds both in the parabolic and elliptic case of the
boundary value problem under the sole hypothesis of probabilistic regularity of
the optimal stopping boundary, thus improving upon known analytic results in
the PDE literature, and establishing the fact for the first time in the case of
integro-differential equations. The method of proof is purely probabilistic and
conceptually simple. Examples of application include the first known
probabilistic proof of the fact that the time derivative of the value function
in the American put problem is continuous across the optimal stopping boundary.
