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Applied Mathematics Research eXpress Advance Access originally published online on June 10, 2009
Applied Mathematics Research eXpress (2009) 2009:1-46, doi:10.1093/amrx/abp001 published on November 6, 2009
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© The Author 2009. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org.

Stochastic Acceleration in an Inhomogeneous Time Random Force Field

Thierry Goudon and Mathias Rousset

Project-Team SIMPAF, INRIA Lille-Nord Europe Research Centre, Park Plazza 40, avenue Halley F-59650 Villeneuve d'Ascq cedex, France
Laboratoire Paul Painlevé CNRS-USTLille, France

Correspondence: Correspondence to be sent to: thierry.goudon{at}inria.fr

This paper studies the asymptotic behavior of a particle with large initial velocity and subject to a force field which is randomly time dependent and inhomogeneous in space. We analyze the diffusive limit Formula of the position–velocity pair under the appropriate space–time rescaling: Formula . Two alternative approaches are proposed. The first one is based on hydrodynamic limits and homogenization techniques for the underlying kinetic equation; the second one is based on homogenization of the random distribution of trajectories. Time randomness is embodied into an underlying Markov process. Space inhomogeneity is modeled by a periodic structure in the first approach, and by a random field in the second one. In the first case, the analysis relies on the dissipation properties of the Markov process, whereas in the second case, the mixing properties of the random field are used. We point out more analogies and differences of the two obtained results.

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