Vol. 3, No. 1, 2021

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Diffusion-approximation in stochastically forced kinetic equations

Arnaud Debussche and Julien Vovelle

Vol. 3 (2021), No. 1, 1–53
Abstract

We derive the hydrodynamic limit of a kinetic equation where the interactions in velocity are modeled by a linear operator (Fokker–Planck or linear Boltzmann) and the force in the Vlasov term is a stochastic process with high amplitude and short-range correlation. In the scales and the regime we consider, the hydrodynamic equation is a scalar second-order stochastic partial differential equation. Compared to the deterministic case, we also observe a phenomenon of enhanced diffusion.

Keywords
diffusion-approximation, kinetic equation, hydrodynamic limit
Mathematical Subject Classification 2010
Primary: 35Q20, 35R60, 60H15, 35B40
Milestones
Received: 15 March 2019
Accepted: 16 November 2019
Published: 20 May 2020
Authors
Arnaud Debussche
Univ Rennes, CNRS, IRMAR - UMR
6625, F-35000 Rennes
France
Julien Vovelle
Univ Lyon
CNRS, ENS Lyon, UMPA - UMR
5669, F-69364 Lyon
France