Open Access
September 2020 Fractional diffusion limit for a kinetic equation with an interface
Tomasz Komorowski, Stefano Olla, Lenya Ryzhik
Ann. Probab. 48(5): 2290-2322 (September 2020). DOI: 10.1214/20-AOP1423


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.


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Tomasz Komorowski. Stefano Olla. Lenya Ryzhik. "Fractional diffusion limit for a kinetic equation with an interface." Ann. Probab. 48 (5) 2290 - 2322, September 2020.


Received: 1 May 2019; Revised: 1 November 2019; Published: September 2020
First available in Project Euclid: 23 September 2020

MathSciNet: MR4152643
Digital Object Identifier: 10.1214/20-AOP1423

Primary: 35Q79 , 45A05 , 60J75

Keywords: boundary conditions at interface , Diffusion limits from kinetic equations , fractional Laplacian , Stable processes

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 5 • September 2020
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