 Ornstein-Uhlenbeck entropy dissipation identity

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 $e^{-2t}\int |h^{\prime }{|}^2/hd\gamma$. Integrating and setting $h=f^2$ proves the Gaussian logarithmic Sobolev inequality. Positive clipped approximation extends it to functions of finite Gaussian Dirichlet energy.