On Feller's boundary problem for Markov processes in weak duality

We give an affirmative answer to Feller's boundary problem going back to 1957 by obtaining a resolvent characterization for the duality preserving extensions of a pair of standard Markov processes in weak duality (minimal processes) to the boundary consisting of countably many points. Our resolvent characterization involves the resolvents for the minimal processes, the Feller measures that are intrinsic to the minimal processes as well as the restrictions to the boundary of the jumping and killing measures of the extension processes. Conversely, given killing rates on the boundary, we construct the corresponding duality preserving extensions of the minimal processes that admit no jumps between the boundary points and have the prescribed killing rate at the boundary, by repeatedly doing one-point extension one at a time using Itô's Poisson point processes of excursions.