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 , 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 ), 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.
@article{SLSEDP_2011-2012____A26_0, author = {Pierre-Louis Lions and Beno{\^\i}t Perthame and Panagiotis E. Souganidis}, title = {Stochastic averaging lemmas for kinetic equations}, journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications}, note = {talk:26}, pages = {1--17}, publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2011-2012}, doi = {10.5802/slsedp.21}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.21/} }
TY - JOUR AU - Pierre-Louis Lions AU - Benoît Perthame AU - Panagiotis E. Souganidis TI - Stochastic averaging lemmas for kinetic equations JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:26 PY - 2011-2012 SP - 1 EP - 17 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.21/ DO - 10.5802/slsedp.21 LA - en ID - SLSEDP_2011-2012____A26_0 ER -
%0 Journal Article %A Pierre-Louis Lions %A Benoît Perthame %A Panagiotis E. Souganidis %T Stochastic averaging lemmas for kinetic equations %J Séminaire Laurent Schwartz — EDP et applications %Z talk:26 %D 2011-2012 %P 1-17 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.21/ %R 10.5802/slsedp.21 %G en %F SLSEDP_2011-2012____A26_0
Pierre-Louis Lions; Benoît Perthame; Panagiotis E. Souganidis. Stochastic averaging lemmas for kinetic equations. Séminaire Laurent Schwartz — EDP et applications (2011-2012), Talk no. 26, 17 p. doi : 10.5802/slsedp.21. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.21/
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