= Mehler formula for the Ornstein-Uhlenbeck semigroup
{c}
{title2=$P_tf(x)=\mathbb E f(e^{-t}x+\sqrt{1-e^{-2t}}Z)$}
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.
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