Flux and lateral conditions for symmetric Markov processes

The purpose of this paper is to give an affirmative answer at infinitesimal generator level to the 40 years old Feller's boundary problem for symmetric Markov processes with general quasi-closed boundaries. For this, we introduce a new notion of flux functional, which can be intrinsically defined via the minimal process X ⁰ in the interior. We then use it to characterize the L ²-infinitesimal generator of a symmetric process that extends X ⁰. Special attention is paid to the case when the boundary consists of countable many points possessing no accumulation points.