  We consider a second order differential operator $\mathscr{A}$ on an
(typically unbounded) open and Dirichlet regular set $\Omega\subset
\mathbb{R}^d$ and subject to nonlocal Dirichlet boundary conditions of the form
\[ u(z) = \int_\Omega u(x)\mu (z, dx) \quad \mbox{ for } z\in \partial \Omega.
\] Here, $\mu : \partial\Omega \to \mathscr{M}(\Omega)$ is a $\sigma
(\mathscr{M}(\Omega), C_b(\Omega))$-continuous map taking values in the
probability measures on $\Omega$. Under suitable assumptions on the
coefficients in $\mathscr{A}$, which may be unbounded, we prove that a
realization $A_\mu$ of $\mathscr{A}$ subject to the nonlocal boundary
condition, generates a (not strongly continuous) semigroup on
$L^\infty(\Omega)$. We also establish a sufficient condition for this semigroup
to be Markovian and prove that in this case, it enjoys the strong Feller
property. We also study the asymptotic behavior of the semigroup.
