In this talk, we will focus on the evolution equations associated with nonselfadjoint quadratic differential operators. The purpose is first to understand how the possible non-commutation phenomena between the selfadjoint and the skew-selfadjoint parts of these operators allow the associated evolution operators to enjoy smoothing and localizing properties in specific directions of the phase space which will be precisely described. These different properties will be deduced from a fine description of the polar decomposition of the evolution operators considered. An application to the generalized Ornstein-Uhlenbeck equations, of which the Kramers-Fokker-Planck equation is a particular case, will be given. We will also explain how a refinement of the aforementioned polar decomposition allows to understand the local smoothing properties and the gains of integrability enjoyed by these equations, under a geometric assumption. These results come from a series of works with J. Bernier (LMJL).
Jérémy Heleine, David Lafontaine
