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  1. In this article, we introduce a model featuring a Lévy process in a bounded domain with semi-transparent boundary, by considering the fractional Laplacian operator with lower order non-local perturbations. We study the wellposedness of the model, certain qualitative properties and Runge type approximation. Furthermore, we consider the inverse problem of determining the unknown coefficients in our model from the exterior measurements of the corresponding Cauchy data. We also discuss the recovery of all the unknown coefficients from a single measurement.