Diffusion with nonlocal boundary conditions

We consider second order differential operators Aμ on a bounded, Dirichlet regular set Ω⊂Rd, subject to the nonlocal boundary conditionsu(z)=∫Ωu(x)μ(z,dx)for z∈∂Ω. Here the function μ:∂Ω→M+(Ω) is σ(M(Ω),Cb(Ω))-continuous with 0≤μ(z,Ω)≤1 for all z∈∂Ω. Under suitable assumptions on the coefficients in Aμ, we prove that Aμ generates a holomorphic positive contraction semigroup Tμ on L∞(Ω). The semigroup Tμ 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(Ω‾). We also prove that Tμ is immediately compact and study the asymptotic behavior of Tμ(t) as t→∞.