We consider the limit of a linear kinetic equation with reflection-transmission-absorption at an interface and with a degenerate scattering kernel. The equation arises from a microscopic chain of oscillators in contact with a heat bath. In the absence of the interface, the solutions exhibit a superdiffusive behavior in the long time limit. With the interface, the long time limit is the unique solution of a version of the fractional in space heat equation with reflection-transmission-absorption at the interface. The limit problem corresponds to a certain stable process that is either absorbed, reflected or transmitted upon crossing the interface.
"Fractional diffusion limit for a kinetic equation with an interface." Ann. Probab. 48 (5) 2290 - 2322, September 2020. https://doi.org/10.1214/20-AOP1423