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We consider dynamic boundary conditions involving non-local operators. Ouranalysis includes a detailed description of such operators together with theirrelations with random times and random (additive) functionals. We provide somenew characterizations for the boundary behaviour of the Brownian motion basedon the interplay between non-local operators and boundary value problems. Ourmain focus is on Feller-Wentzell diffusions. We first consider the instructivecase of the real line, then we extend our results on star graphs with trappingpoints or repulsive vertices.

Non-local Boundary Value Problems, stochastic resetting and Brownian motions on graphs