= Ornstein-Uhlenbeck entropy dissipation identity
{c}
{title2=$-\tfrac d{dt}\operatorname{Ent}_\gamma(P_th)=\int|(P_th)'|^2/(P_th)\,d\gamma$}
For bounded smooth positive $h$ bounded away from zero, differentiation and <Gaussian integration by parts> give the displayed identity. The <Ornstein-Uhlenbeck gradient commutation identity> and weighted <Cauchy-Schwarz inequality> bound its right side by $e^{-2t}\int|h'|^2/h\,d\gamma$. Integrating and setting $h=f^2$ proves the <Gaussian logarithmic Sobolev inequality>. Positive clipped approximation extends it to functions of finite <Gaussian Dirichlet energy>.
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