Smoothing properties of semigroups generated by fractional Ornstein-Uhlenbeck operators

The Ornstein-Uhlenbeck operators are non-local and non-selfadjoint operators defined as the sum of a fractional diffusion and a linear transport part. Due to the non-commuatation between the selfadjoint and skew-selfadjoint parts of these operators, partial hypoellipticity phenomena can occur that allow their associated semigroups to enjoy some smoothing properties in specific directions of the phase space. The aim of this talk is to present and explain these phenomena. Some applications to the null-controllability of fractional Ornstein-Uhlenbeck equations posed on the whole Euclidean space will also be given.