= Ornstein-Uhlenbeck gradient commutation identity
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{title2=$(P_tf)'=e^{-t}P_t(f')$}
Differentiating the <Mehler formula for the Ornstein-Uhlenbeck semigroup> proves the identity for continuously differentiable functions with bounded <derivative>. Density extends it to the Gaussian Sobolev form domain. Together with a weighted <Cauchy-Schwarz inequality>, it yields the <Ornstein-Uhlenbeck entropy dissipation identity> proof of the <Gaussian logarithmic Sobolev inequality>.
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