  We investigate in a systematic way hypercontractivity property in Orlicz
spaces for Markov semi-groups related to homogeneous and non homogeneous
diffusions in $\mathbb{R}^n$. We provide an explicit construction of a family
of Orlicz functions for which we prove that the associated hypercontractivity
property is equivalent to a suitable functional inequality.
  Connections between our results and the Talagrand Conjecture on the
regularization effect of the Ornstein-Uhlenbeck semi-group in $\mathbb{L}_1$
are also discussed.
