Stochastic averaging lemmas for kinetic equations

Pierre-Louis Lions[1]; Benoît Perthame[2]; Panagiotis E. Souganidis[3]

  • [1] Université Paris-Dauphine place du Maréchal-de-Lattre-de-Tassigny 75775 Paris cedex 16 France
  • [2] Université Pierre et Marie Curie, Paris 06 CNRS UMR 7598 Laboratoire J.-L. Lions, BC187 4, place Jussieu F-75252 Paris 5 and INRIA Paris-Rocquencourt EPI Bang
  • [3] Department of Mathematics University of Chicago Chicago, IL 60637, USA

Séminaire Laurent Schwartz — EDP et applications (2011-2012)

  • Volume: 2011-2012, page 1-17
  • ISSN: 2266-0607

Abstract

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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.

How to cite

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Lions, Pierre-Louis, Perthame, Benoît, and Souganidis, Panagiotis E.. "Stochastic averaging lemmas for kinetic equations." Séminaire Laurent Schwartz — EDP et applications 2011-2012 (2011-2012): 1-17. <http://eudml.org/doc/251165>.

@article{Lions2011-2012,
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.},
affiliation = {Université Paris-Dauphine place du Maréchal-de-Lattre-de-Tassigny 75775 Paris cedex 16 France; Université Pierre et Marie Curie, Paris 06 CNRS UMR 7598 Laboratoire J.-L. Lions, BC187 4, place Jussieu F-75252 Paris 5 and INRIA Paris-Rocquencourt EPI Bang; Department of Mathematics University of Chicago Chicago, IL 60637, USA},
author = {Lions, Pierre-Louis, Perthame, Benoît, Souganidis, Panagiotis E.},
journal = {Séminaire Laurent Schwartz — EDP et applications},
keywords = {stochastic kinetic equations; stochastic conservation laws; averaging lemmas; fractional Sobolev spaces},
language = {eng},
pages = {1-17},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Stochastic averaging lemmas for kinetic equations},
url = {http://eudml.org/doc/251165},
volume = {2011-2012},
year = {2011-2012},
}

TY - JOUR
AU - Lions, Pierre-Louis
AU - Perthame, Benoît
AU - Souganidis, Panagiotis E.
TI - Stochastic averaging lemmas for kinetic equations
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2011-2012
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2011-2012
SP - 1
EP - 17
AB - 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.
LA - eng
KW - stochastic kinetic equations; stochastic conservation laws; averaging lemmas; fractional Sobolev spaces
UR - http://eudml.org/doc/251165
ER -

References

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