  We consider dynamic boundary conditions involving non-local operators. Our
analysis includes a detailed description of such operators together with their
relations with random times and random (additive) functionals. We provide some
new characterizations for the boundary behaviour of the Brownian motion based
on the interplay between non-local operators and boundary value problems.
Although our main focus is about Feller diffusions on bounded domains whose
boundary is a set of positive Lebesgue measure, due to the novelty of such
results we present here a first study concerned with the motions on the
positive real line. In our view, also for the real line, the results appears
stimulating and they anticipate interesting applications.
