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

TL;DR

This paper investigates hypercontractivity and Log-Sobolev inequalities for Ornstein-Uhlenbeck operators in infinite-dimensional spaces, with applications to stochastic PDEs, advancing understanding of their functional inequalities and evolution properties.

Contribution

It establishes hypercontractivity results for Ornstein-Uhlenbeck operators in infinite dimensions using Log-Sobolev inequalities, including applications to non-autonomous stochastic PDEs.

Findings

Proves hypercontractivity for Ornstein-Uhlenbeck operators in infinite dimensions.

Derives Log-Sobolev inequalities for evolution operators.

Applies results to stochastic parabolic PDEs.

Abstract

In an infinite dimensional separable Hilbert space $X$, we study the realizations of Ornstein-Uhlenbeck evolution operators $\pst$ in the spaces $L^p(X,\g_t)$, $\{\g_t\}_{t\in\R}$ being the unique evolution system of measures for $\pst$ in $\R$. 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.