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Fractional diffusion limit for a kinetic equation with an interface

Komorowski, Tomasz; Olla, Stefano; Ryzhik, Lenya (2020), Fractional diffusion limit for a kinetic equation with an interface, Annals of Probability, 48, 5, p. 2290-2322. 10.1214/20-AOP1423

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Type
Article accepté pour publication ou publié
Date
2020
Journal name
Annals of Probability
Volume
48
Number
5
Publisher
Institute of Mathematical Statistics
Published in
Paris
Pages
2290-2322
Publication identifier
Metadata
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Author(s)
Komorowski, Tomasz
Instytut Matematyki = Institute of Mathematics [Lublin]
Olla, Stefano cc

Ryzhik, Lenya
Abstract (EN)
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.
Subjects / Keywords
Diffusion Limits from Kinetic Equations; Fractional Laplacian; Stable Processes; Boundary Conditions at Interface