Diffusion with nonlocal boundary conditions

We consider second order differential operators A(mu) on a bounded, Dirichlet regular set Omega subset of R-d, subject to the nonlocal boundary conditions u(z) = integral(Omega) u(x) mu(z,dx) for z is an element of partial derivative Omega. Here the function mu : partial derivative Omega -> M+ (Omega) is sigma(M(Omega), C-b(n Omega))-continuous with 0 <= mu(z, Omega) <= 1 for all z is an element of partial derivative 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-infinity(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((Omega) over bar). We also prove that T-mu is immediately compact and study the asymptotic behavior of T-mu (t) as t -> infinity. (C) 2016 Published by Elsevier Inc.