Thermal boundaries for energy superdiffusion

We consider a chain of oscillators with one particle in contact with a thermostat at temperature T. The thermostat is modeled by a Langevin dynamics or a renewal of the velocity with a gaussian random variable with variance T. The dynamics of the oscillators is perturbed by a random exchange on velocities between nearest neighbor particles.
The (thermal) energy has a macroscopic superdiffusive behavior governed by a fractional heat equation (i.e. with a fractional Laplacian). The microscopic thermostat impose a particular boundary condition to the fractional Laplacian, corresponding to certain probabilities of transmission/reflection/absorption/creation for the corresponding superdiffusive Levy process.
This is from a series of works in collaboration with Tomazs Komorowski, Lenya Ryzhik, Herbert Spohn.