= Ornstein-Uhlenbeck semigroup
{c}
{title2=$P_t=e^{t(D^2-xD)}$}
The one-dimensional semigroup associated with the standard <Ornstein-Uhlenbeck process> has invariant standard <Gaussian measure> and generator $Lf=f''-xf'$. Its <Mehler formula for the Ornstein-Uhlenbeck semigroup> makes <positivity> and invariance explicit. The normalized <Probabilists' Hermite polynomials> diagonalize it, with $P_th_n=e^{-nt}h_n$. It has <spectral gap> one and satisfies the sharp <Gaussian Poincaré inequality>.
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