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

18 pages, no figures; comments of the referees incorporated