Characterization of Gaussian semigroups on separable Banach spaces (Q1880369)

Let \(E\) be a separable real Banach space and denote by \(\text{BUC}(E)\) the space of bounded and uniformly continuous functions on \(E\). Let \(\mu\) be a Gaussian measure defined on the Borel \(\sigma\)-algebra of \(E\). A \(C_0\)-semigroup \((T(t))_{t\geq 0}\) acting on \(\text{BUC}(E)\) is called Gaussian if \[ (T(t)f)(x) = \int_E f(x+\sqrt{t}y)d\mu(y),\;t \geq 0,\;f\in \text{BUC}(E). \] The authors obtained necessary and sufficient conditions ensuring that \((T(t))_{t\geq 0}\) is Gaussian. This characterization involves, among other conditions, the locality of the generator of \((T(t))_{t\geq 0}\) and a certain family of cylindrical functions.