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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.

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Dr. Khalid Akhlil is an Assistant professor of mathematics in the Polydisciplinary Faculty of Ouarzazate, Morocco. He started his doctoral researches in Agadir (Morocco) and then earned a German fellowship for a long research stay in Ulm. His main subjects of research are functional analysis, stochastic processes and fuzzy structures.

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