We consider second order differential operators $A_��$ on a bounded, Dirichlet regular set $��\subset \mathbb{R}^d$, subject to the nonlocal boundary conditions \[ u(z) = \int_��u(x)\, ��(z, dx)\quad \mbox{for } z \in \partial ��. \] Here the function $��: \partial��\to \mathscr{M}^+(��)$ is $��(\mathscr{M} (��), C_b(��))$-continuous with $0\leq ��(z,��) \leq 1$ for all $z\in \partial ��$. Under suitable assumptions on the coefficients in $A_��$, we prove that $A_��$ generates a holomorphic positive contraction semigroup $T_��$ on $L^\infty(��)$. The semigroup $T_��$ is never strongly continuous, but it enjoys the strong Feller property in the sense that it consists of kernel operators and takes values in $C(\bar��)$. We also prove that $T_��$ is immediately compact and study the asymptotic behavior of $T_��(t)$ as $t \to \infty$.

18 pages, no figures; comments of the referees incorporated