Title: Boundary problems for the fractional and tempered fractional operators,
Abstract: For characterizing the Brownian motion in a bounded domain: $\Omega$, it is
well-known that the boundary conditions of the classical diffusion equation
just rely on the given information of the solution along the boundary of a
domain; on the contrary, for the Lévy flights or tempered Lévy flights in a
bounded domain, it involves the information of a solution in the complementary
set of $\Omega$, i.e., $\mathbb{R}^n\backslash \Omega$, with the potential
reason that paths of the corresponding stochastic process are discontinuous.
Guided by probability intuitions and the stochastic perspectives of anomalous
diffusion, we show the reasonable ways, ensuring the clear physical meaning and
well-posedness of the partial differential equations (PDEs), of specifying
`boundary' conditions for space fractional PDEs modeling the anomalous
diffusion. Some properties of the operators are discussed, and the
well-posednesses of the PDEs with generalized boundary conditions are proved.