To characterize the Brownian motion in a bounded domain, it is well known that the boundary conditions of the classical di usion equation just rely on the given information of the solution along the boundary of a domain; in contrast, for the L evy ights or tempered L evyights in a bounded domain, the boundary conditions involve the information of a solution in the complementary set of, i.e., Rnn, with the potential reason that paths of the corresponding stochastic process are discontinuous. Guided by probability intuitions and the stochastic perspectives of anomalous di usion, we show the reasonable ways, ensuring the clear physical meaning and well-posedness of the partial di erential equations (PDEs), of specifying boundary" conditions for space fractional PDEs modeling the anomalous di usion. Some properties of the operators are discussed, and the well-posednesses of the PDEs with generalized boundary conditions are proved.