 Solution

The given relative Fisher information inequality is $I^{\prime }(t)\le -2I(t)$. Multiplying by $e^{2t}$ and integrating gives relative Fisher information decay under Ornstein-Uhlenbeck flow:In particular $I(t)\le I(0)e^{-2t}$ when $I(0)$ is finite; if necessary one starts at a positive time with finite information. It follows that $I(t)\to 0$.

Write $H(t)=H(f_t\mid \gamma )$. It is nonnegative and decreasing, so has a finite limit $H_{\infty }\ge 0$ when $H(0)<\infty$. The printed integrability request concerns the product $H(t)H^{\prime }(t)$. Its sign is nonpositive, and the fundamental theorem of calculus givesPassing to $R\to \infty$ proves time integrability of an entropy-dissipation product:Also ${\int }_0^{\infty }|H^{\prime }|=H(0)-H_{\infty }\le H(0)$. The zero value of $H_{\infty }$ will be used as supplied in (f); positivity and monotonicity alone only establish existence of the limit.