Mehler formula for the Ornstein-Uhlenbeck semigroup ($P_{t}f(x)=\mathbb{E}f(e^{-t}x+\sqrt{1-e^{-2t}}Z)$)

Here $Z$ has the standard normal distribution. Applied to the generating function $e^{sx-s^2/2}$, the Gaussian moment-generating function replaces $s$ by $e^{-t}s$, proving the Hermite eigenvalue identity. Positivity and preservation of Gaussian measure then give the $L^2$ contraction and identify the full Ornstein-Uhlenbeck semigroup by polynomial density.