Hypercontractivity and Absolute Continuity of Gaussian Generalized Mehler Semigroups in Hilbert Spaces

Abstract

In this paper, we prove the hypercontractivity of a non-differentiable Gaussian generalized Mehler semigroup using direct probabilistic arguments. This result implies the exponential convergence of the semigroup at infinity. Under some additional hypotheses, we also establish the absolute continuity of the semigroup with respect to its invariant measure.