Stochastic averaging lemmas for kinetic equations

Pierre Louis Lions (1, 2) , Benoît Perthame (3, 4) , Panagiotis E. Souganidis (5)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Collège de France - Chaire Équations aux dérivées partielles et applications
3 LJLL - Laboratoire Jacques-Louis Lions
4 BANG - Nonlinear Analysis for Biology and Geophysical flows
5 Department of Mathematics [Chicago]

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.

Mots clés: Stochastic kinetic equations; stochastic conservation laws; averaging lemmas; fractional Sobolev spaces.; fractional Sobolev spaces

Domaines: Equations aux dérivées partielles [math.AP]

Pierre Louis Lions, Benoît Perthame, Panagiotis E. Souganidis. Stochastic averaging lemmas for kinetic equations. Stochastic averaging lemmas for kinetic equations, Jan 2012, Palaiseau, France. 17pp. ⟨hal-00684372⟩