Stochastic averaging lemmas for kinetic equations
Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 26, 17 p.

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.

@article{SLSEDP_2011-2012____A26_0,
author = {Lions, Pierre-Louis and Perthame, Beno{\^\i}t and Souganidis, Panagiotis E.},
title = {Stochastic averaging lemmas for kinetic equations},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:26},
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 = {http://archive.numdam.org/articles/10.5802/slsedp.21/}
}
Lions, Pierre-Louis; Perthame, Benoît; Souganidis, Panagiotis E. Stochastic averaging lemmas for kinetic equations. Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 26, 17 p. doi : 10.5802/slsedp.21. http://archive.numdam.org/articles/10.5802/slsedp.21/

[1] C. Bardos, P. Degond, Global existence for the Vlasov-Poisson equation in $3$ space variables with small initial data. Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 2, 101–118. | Numdam | MR 794002 | Zbl 0593.35076

[2] F. Bouchut: Introduction to the mathematical theory of kinetic equations, in Kinetic Equations and Asymptotic Theories, (F. Bouchut, F. Golse and M. Pulvirenti), Series in Appl. Math. no. 4, Elsevier (2000). | Zbl 0979.82048

[3] F. Bouchut and L. Desvillettes Averaging lemmas without time Fourier transform and application to discretized kinetic equations, Proceedings of the Royal Society of Edinburgh, 129A, 19-36 (1999). | MR 1669221 | Zbl 0933.35159

[4] A. Debussche and J. Vovelle, Scalar conservation laws with stochastic forcing. To appear in Communications on Pure and Applied Analysis. | MR 2652180 | Zbl 1200.60050

[5] A. Debussche and J. Vovelle, Diffusion limit for a stochastic kinetic problem. J. Funct. Anal. 259 (2010), no. 4, 1014–1042. | MR 2652180 | Zbl 1200.60050

[6] R.J. DiPerna, P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems. Comm. Pure Appl. Math. 42, p. 729–757, (1989) | MR 1003433 | Zbl 0698.35128

[7] R.J. DiPerna, P.-L. Lions, On the Cauchy problem for the Boltzmann equation: global existence and weak stability results. Annals of Math 130 (1990) 321–366. | MR 1014927 | Zbl 0698.45010

[8] R.J. DiPerna, P.-L. Lions and Y. Meyer, ${L}^{p}$ regularity of velocity averages, Ann. Inst. H. Poincaré, Analyse non-linéaire 8(3–4) (1991) 271–287. | Numdam | MR 1127927 | Zbl 0763.35014

[9] F. Flandoli, Random Perturbation of PDEs and Fluid Dynamic Models, Lecture Notes in Mathematics 2015, Springer 2011 | MR 2796837 | Zbl 1221.35004

[10] F. Flandoli, The Interaction Between Noise and Transport Mechanisms in PDEs. Milan J. Math. Vol. 79 (2011) 543–560. | MR 2862027 | Zbl 1231.35165

[11] P. Gérard, Moyennisation et régularité deux-microlocale. (French) [Second-microlocal averaging and regularity] Ann. Sci. École Norm. Sup. (4) 23 (1990), no. 1, 89–121. | Numdam | MR 1042388 | Zbl 0725.35003

[12] R. T. Glassey, The Cauchy problem in kinetic theory, SIAM publications, Philadelphia (1996). | MR 1379589 | Zbl 0858.76001

[13] F. Golse, B. Perthame and R. Sentis, Un résultat de compacité pour les équations du transport..., C. R. Acad. Sci. Paris, Série I 301 (1985) 341–344. | MR 808622 | Zbl 0591.45007

[14] F. Golse, P.-L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 76 (1988) 110–125. | MR 923047 | Zbl 0652.47031

[15] P.-L. Lions, B. Perthame and P. E. Souganidis, Scalar conservation laws with rough (stochastic) fluxes. Work in preparation.

[16] P.-L. Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc., 7 (1) 169–191 (1994). | MR 1201239 | Zbl 0820.35094

[17] P.-L. Lions and P. E. Souganidis, Équations aux dérivées partielles stochastiques nonlinéaires et solutions de viscosité, Seminaire: Équations aux Dérivées Partielles, 1998–1999, Exp. No. I, 15, École Polytech., Palaiseau, 1999. | MR 1721319 | Zbl 1061.35530

[18] P.-L. Lions and P. E. Souganidis, Viscosity solutions of fully nonlinear stochastic partial differential equations, Viscosity solutions of differential equations and related topics (Japanese) Kyoto, 2001, Sūrikaisekikenkyūsho Kōkyūroku, 1287, 2002, 58–65. | MR 1959710

[19] B. Perthame, Mathematical Tools for Kinetic Equations. Bull. Amer. Math. Soc. 41, 205–244 (2004). | MR 2043752 | Zbl 1151.82351

[20] B. Perthame and P.E. Souganidis, A limiting case of velocity averaging lemmas Ann. Sc. E.N.S. Série 4, t. 31 (1998) 591–598. | Numdam | MR 1634024 | Zbl 0956.45010

[21] E. M. Stein. Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton (1970). | MR 290095 | Zbl 0207.13501

[22] C. Villani, A review of mathematical topics in collisional kinetic theory.Ê Handbook of mathematical fluid dynamics, Vol. I,Ê 71–305, North-Holland, Amsterdam, 2002. | MR 1942465 | Zbl 1170.82369