Log-Sobolev inequalities and hypercontractivity for Ornstein-Uhlenbeck evolution operators in infinite dimensions

In an infinite dimensional separable Hilbert space $X$, we study the realizations of Ornstein-Uhlenbeck evolution operators $\text{pst}$ in the spaces $L^p(X,{\text{g}}_t)$, $\{{\text{g}}_t{\}}_{t\in \mathbb{R}}$ being the unique evolution system of measures for $\text{pst}$ in $\mathbb{R}$. We prove hyperconctractivity results, relying on suitable Log-Sobolev estimates. Among the examples we consider the transition evolution operator of a non autonomous stochastic parabolic PDE.