Local and Nonlocal Robin Boundary Value Problems

Using a capacity approach and the theory of measure''s perturbation of Dirichlet forms we give the probabilistic representation of the General Robin boundary value problems on an arbitrary domain. In the first time we study the case of positive smooth measures and then we focus on signed smooth measures case by defining a special Kato class of measures. Some other topics related to the first one are also studied. In particular we aim to characterize all semi-groups sandwiched between Dirichlet and Neumann ones. In addition we characterize the closability of a general non local Robin boundary value problem by defining a new notion of admissibility and we study the positivity of the non-local Robin Laplacian involving bounded operators.