Wave packet analysis of semigroups generated by quadratic differential operators

We perform a phase space analysis of evolution equations associated with the Weyl quantization qw of a complex quadratic form q on R2d with non-positive real part. In particular, we obtain pointwise bounds for the matrix coefficients of the Gabor wave packet decomposition of the generated semigroup e^{tq^w} if Req<=0 and the companion singular space associated is trivial. This result is then leveraged to achieve a comprehensive analysis of the phase space regularity of e^{tq^w} with Req<=0, thereby extending the L2 analysis of quadratic semigroups initiated by Hitrik and Pravda-Starov to general modulation spaces Mp(Rd), 1<=p<=infty, with optimal explicit bounds.