Ornstein–Uhlenbeck semigroups in infinite dimension

Abstract

This is a survey paper about Ornstein–Uhlenbeck semigroups in infinite dimension and their generators. We start from the classical Ornstein–Uhlenbeck semigroup on Wiener spaces and then discuss the general case in Hilbert spaces. Finally, we present some results for Ornstein–Uhlenbeck semigroups on Banach spaces. This article is part of the theme issue ‘Semigroup applications everywhere’.

Improved Bound of Four Moment Theorem and Its Application to Orthogonal Polynomials Associated with Laws

“…This product Formula (22) gives that the upper chaos grade and lower chaos grade of H q are u = 2 and e = 2 q ≤ 1 for q ≥ 2. Hence, Theorem 3 yields that…”

Time regularity for generalized Mehler semigroups

“…\end{equation}$$The (easy) proof of this statement may be found e.g. in [25, §2(c)]. From (3.1) we get immediately $$$$\begin{equation*} \Vert R(\lambda , L)\Vert _{\mathcal {L}(C_b(X))} \le \frac{M}{\lambda - \beta }, \quad \lambda >\beta .$$…”

“…The theory of Ornstein–Uhlenbeck operators and semigroups in infinite dimensional spaces is very rich; we refer to the book [12] for the basic theory in Hilbert spaces, to the survey paper [17] for the theory in Banach spaces, and to the more recent survey paper [25] for an up‐to‐date account.…”