Feller semigroups and degenerate elliptic operators III

This paper is devoted to the functional analytic approach to the problem of construction of Feller semigroups in the characteristic case via the Fichera function. Probabilistically, our result may be stated as follows: We construct a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves continuously in the interior of the state space, without reaching the boundary. We make use of the Hille–Yosida–Ray theorem that is a Feller semigroup version of the classical Hille–Yosida theorem in terms of the positive maximum principle. Our proof is based on a method of elliptic regularizations essentially due to Oleĭnik and Radkevič. The weak convergence of approximate solutions follows from the local sequential weak compactness of Hilbert spaces and Mazur's theorem in normed linear spaces.