= Solution
The given <relative Fisher information> inequality is $I'(t)\leq-2I(t)$. Multiplying by $e^{2t}$ and integrating gives \b[<relative Fisher information decay under Ornstein-Uhlenbeck flow>]:
$$
\boxed{I(f_t\mid\gamma)\leq e^{-2(t-s)}I(f_s\mid\gamma)
\qquad(t\geq s\geq0).}
$$
In particular $I(t)\leq I(0)e^{-2t}$ when $I(0)$ is finite; if necessary one starts at a positive time with finite information. It follows that $I(t)\to0$.
Write $H(t)=H(f_t\mid\gamma)$. It is nonnegative and decreasing, so has a finite limit $H_\infty\geq0$ when $H(0)<\infty$. The printed integrability request concerns the product $H(t)H'(t)$. Its sign is nonpositive, and the <fundamental theorem of calculus> gives
$$
\int_0^R|H(t)H'(t)|\,dt
=-\frac12\int_0^R(H(t)^2)'\,dt
=\frac{H(0)^2-H(R)^2}{2}.
$$
Passing to $R\to\infty$ proves \b[<time integrability of an entropy-dissipation product>]:
$$
\boxed{H(t)H'(t)\in L^1(0,\infty),\qquad
\|HH'\|_{L^1}=\frac{H(0)^2-H_\infty^2}{2}\leq\frac{H(0)^2}{2}.}
$$
Also $\int_0^\infty|H'|=H(0)-H_\infty\leq H(0)$. The zero value of $H_\infty$ will be used as supplied in (f); positivity and monotonicity alone only establish existence of the limit.
Back to article page
