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

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