Diffusion with nonlocal boundary conditions

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