  We extend the results obtained in \cite{Dov22} by introducing a new class of
boundary value problems involving non-local dynamic boundary conditions. We
focus on the problem to find a solution to a local problem on a domain $\Omega$
with non-local dynamic conditions on the boundary $\partial \Omega$. Due to the
pioneering nature of the present research, we propose here the apparently
simple case of $\Omega=(0, \infty)$ with boundary $\{0\}$ of zero Lebesgue
measure. Our results turn out to be instructive for the general case of
boundary with positive (finite) Borel measures. Moreover, in our view, we bring
new light to dynamic boundary value problems and the probabilistic description
of the associated models.
