Psi-entropy inequalities for the Ornstein-Uhlenbeck semigroup

We study the Ornstein-Uhlenbeck semigroup ( P t ) t ≥0 ={exp( tL )} t ≥0 generated by the operator Lf ( x )=Δ f ( x )− x ⋅∇ f ( x ), on ℝ n equipped with the n -dimensional standard Gaussian measure . By means of a simple method involving essentially a commutation property between the semigroup and the gradient, we describe a large family of optimal integral inequalities with the logarithmic Sobolev and reverse logarithmic Sobolev inequalities as particular cases.