# Stochastic averaging lemmas for kinetic equations

3 BANG - Nonlinear Analysis for Biology and Geophysical flows
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt
Abstract : We develop a class of averaging lemmas for stochastic kinetic equations. The velocity is multiplied by a white noise which produces a remarkable change in time scale. Compared to the deterministic case and as far as we work in $L^2$, the nature of regularity on averages is not changed in this stochastic kinetic equation and stays in the range of fractional Sobolev spaces at the price of an additional expectation. However all the exponents are changed; either time decay rates are slower (when the right hand side belongs to $L^2$), or regularity is better when the right hand side contains derivatives. These changes originate from a different space/time scaling in the deterministic and stochastic cases. Our motivation comes from scalar conservation laws with stochastic fluxes where the structure under consideration arises naturally through the kinetic formulation of scalar conservation laws.
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Communication dans un congrès
Stochastic averaging lemmas for kinetic equations, Jan 2012, Palaiseau, France. cedram.org, 2011-2012 (Exp No 26), 17pp, 2011

https://hal.archives-ouvertes.fr/hal-00684372
Contributeur : Benoît Perthame <>
Soumis le : dimanche 1 avril 2012 - 19:35:46
Dernière modification le : mercredi 31 mai 2017 - 01:09:48
Document(s) archivé(s) le : lundi 2 juillet 2012 - 03:19:49

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• HAL Id : hal-00684372, version 1
• ARXIV : 1204.0317

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Pierre-Louis Lions, Benoît Perthame, Panagiotis E. Souganidis. Stochastic averaging lemmas for kinetic equations. Stochastic averaging lemmas for kinetic equations, Jan 2012, Palaiseau, France. cedram.org, 2011-2012 (Exp No 26), 17pp, 2011. <hal-00684372>

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