Fractional diffusion limit for a kinetic equation with an interface

2019/05/25 by Komorowski, Tomasz, Olla, Stefano, Ryzhik, Lenya · 1 citation
#35Q82 #35R11 #60F17 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1905.10586

Abstract

We consider the limit of a linear kinetic equation, with reflection-transmission-absorption at an interface, with a degenerate scattering kernel. The equation arise 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.

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• Stochastic oscillators out of equilibrium: scaling limits and correlation estimates

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