Non-local Boundary Value Problems for Brownian motions

We consider dynamic boundary conditions involving non-local operators. Our analysis includes a detailed description of such operators together with their relations with random times and random (additive) functionals. We provide some new characterizations for the boundary behaviour of the Brownian motion based on the interplay between non-local operators and boundary value problems. Although our main focus is about Feller diffusions on bounded domains whose boundary is a set of positive Lebesgue measure, due to the novelty of such results we present here a first study concerned with the motions on the positive real line. In our view, also for the real line, the results appears stimulating and they anticipate interesting applications.