A unification of hypercontractivities of the Ornstein-Uhlenbeck semigroup and its connection with Phi-entropy inequalities

摘要

Let gamma(d) be the d-dimensional standard Gaussian measure and Q = {Q(t)}(t = 0) the Ornstein-Uhlenbeck semigroup acting on L-1(gamma(d)). The semigroup Q enjoys the hypercontractivity, which is also known to be equivalent to the exponential hypercontractivity. In this paper, by employing stochastic analysis, we derive a family of inequalities that unifies the exponential and original hypercontractivities; a generalization of the Gaussian logarithmic Sobolev inequality is obtained as a corollary. We then discuss a connection of those results with Phi-entropy inequalities in a general framework of Markov semigroups. A unification of the exponential hypercontractivity and the reverse hypercontractivity of the Ornstein-Uhlenbeck semigroup Q is also provided. (C) 2018 Elsevier Inc. All rights reserved.