Ornstein-Uhlenbeck semigroup and Fourier transform acting on positive finite measures on the Schwartz space (Q1912817)

Let \(A=1+u^2\cdot-{d^2\over du^2}\); a generalized Ornstein-Uhlenbeck semigroup on \(L^2({\mathcal S}^*,\mu)\) is defined as \(T^A_tF(x)= \int_{{\mathcal S}^*} F(e^{-tA} x+ \sqrt{1-e^{-2tA}} y)\mu(dy)\), where \(\mu\) is the standard Gaussian measure on the Schwartz space \({\mathcal S}^*\). For any bounded \({\mathcal S}^*\)-continuous function \(F\), the above integral can be expressed by an integration with a continuous kernel \(D_t: {\mathcal S}^* \times {\mathcal S}^* \to R: (T^A_tF) (x)= \int F(y) D_t(x,y) \mu(dy)\). This expression is extended to a positive finite measure \(\nu\) on \({\mathcal S}^*\), \(T^A_t \nu(x)= \int D_t(x,y) \nu(dy)\). The results are \(T^A_t \nu(x) \mu (dx) \to\nu (dx)\) weakly when \(t\to\infty\), and \(T^A_t \nu(x)\to {d\nu\over d\mu} (x)\), \(\mu\) a.e. \(t\to\infty\). Define a Fourier transform of \(\nu\) as \(F[\nu](x)= e^{|x|^2/2} \int e^{\sqrt {-1} (x,y)} \nu(dy)\), then the O-U semigroup is approximated by an \(L_1\) limit of the Fourier images of finite-dimensional projections related to the eigenvectors of \(A\).