Diffusion with nonlocal Dirichlet boundary conditions on unbounded domains

We consider a second order differential operator $\mathcal{A}$ on an (typically unbounded) open and Dirichlet regular set $\Omega\subset $\mathbb{R}$^d$ and subject to nonlocal Dirichlet boundary conditions of the form $$u(z)={\int }_{\mathrm{\Omega }}u(x)\mu (z,dx)forz\in \mathrm{\partial }\mathrm{\Omega }\mathrm{.}$$ Here, $\mu :\mathrm{\partial }\mathrm{\Omega }\to \mathcal{M}(\mathrm{\Omega })$ is a $\sigma (\mathscr{M}(\Omega), C_b(\Omega))$-continuous map taking values in the probability measures on $\mathrm{\Omega }$. Under suitable assumptions on the coefficients in $\mathcal{A}$, which may be unbounded, we prove that a realization $A_{\mu }$ of $\mathcal{A}$ subject to the nonlocal boundary condition, generates a (not strongly continuous) semigroup on $L^{\mathrm{\infty }}(\mathrm{\Omega })$. 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.