Diffusion with nonlocal Dirichlet boundary conditions on unbounded domains

We consider a second order differential operator A on an open and Dirichlet regular set Omega subset of R-d (typically unbounded) and subject to nonlocal Dirichlet boundary conditions of the form u(z) = integral(Omega) u(x) mu(z, dx) for z is an element of partial derivative Omega. Here, mu : partial derivative Omega -> M(Omega) takes values in the probability measures on Omega and is continuous in the weak topology sigma(M(Omega), C-b(Omega)). Under suitable assumptions on the coefficients of A, which may be unbounded, we prove that a realization A(mu) of A subject to the above nonlocal boundary condition generates a (not strongly continuous) semigroup on L-infinity(Omega). We 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.