Continuity and representation of Gaussian Mehler semigroups
Let \(E\) be a real Banach space and let \(\mu= \{\mu_t\}_{t\geq 0}\) be a one-parameter family of probability measures defined on the \(\sigma\)-algebra \(\Sigma\) generated by the cylindrical subsets of \(E\), i.e., sets \(V\) of the form \[ V= \{x\in E: (\langle x,x^*_1\rangle,\dots, \langle x,x^*_n\rangle)\in B\}, \] where \(x^*_1,\dots, x^*_n\in E\) and \(B\subset \mathbb{R}^n\) is a Borel set. Let \(S= \{S(t)\}_{t\geq 0}\) be a strongly continuous semigroup of linear operators (briefly, a \(C_0\)-semigroup) on \(E\). The pair \((S,\mu)\) is called a Mehler semigroup on \(E\) if in \(\Sigma\) we have \[ \mu_{t+2}= (S(s)\mu_t)* \mu_s,\quad t,s\geq 0, \] where \(S(s)\mu_t\) denotes the image measure of \(\mu_t\) under \(S(s)\). In this paper, the author proves the following result: Let \(\{S(t)\}_{t\geq 0}\) be a \(C_0\)-semigroup on a real Banach space \(E\), and let \(\{Q_t\}_{t\geq 0}\) be a family of positive symmetric operators in \({\mathcal L}(E,E^*)\) such that \(Q_{t+s}= Q_s+ S^*(s) Q_t S(s)\) for all \(t\geq 0\) and \(s\geq 0\). Then the following assertions are equivalent: (i) \(\limsup_{t\downarrow 0}(1/t)\|Q_t\|< \infty\); (ii) there exists a positive symmetric operator \(Q\in{\mathcal L}(E, E^*)\) such that \[ Q_tx=\int^t_0 S^*(s) QS(s) x ds,\quad t\geq 0,\quad x\in E. \] In this situation, \(Q\) is the unique operator with these properties.
continuous semigroup of linear operators
scientific article; zbMATH DE number 1569534