On the existence of Feller semigroups with Dirichlet condition

This paper is devoted to the functional analytic approach to the problem of construction of Feller semigroups with Dirichlet boundary condition in the $characteristic$ case. Intuitively, our result may be stated as follows: One can construct a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves continuously in the state space until it "$dies$" at which time it reaches the set where the absorption phenomenon occurs.