 Ornstein-Uhlenbeck semigroup

The one-dimensional semigroup associated with the standard Ornstein-Uhlenbeck process has invariant standard Gaussian measure and generator $Lf=f^{\prime \prime }-x{f}^{\prime }$. Its Mehler formula for the Ornstein-Uhlenbeck semigroup makes positivity and invariance explicit. The normalized Probabilists' Hermite polynomials diagonalize it, with $P_t{h}_n=e^{-nt}{h}_n$. It has spectral gap one and satisfies the sharp Gaussian Poincaré inequality.