= Solution
Use the zero entropy limit supplied in this subpart. By the <fundamental theorem of calculus> and the <entropy dissipation identity for Ornstein-Uhlenbeck flow>,
$$
H(f_t\mid\gamma)=\int_t^\infty I(f_s\mid\gamma)\,ds.
$$
Apply the <relative Fisher information> decay estimate starting at time $t$:
$$
H(f_t\mid\gamma)\leq I(f_t\mid\gamma)
\int_t^\infty e^{-2(s-t)}\,ds=\frac12I(f_t\mid\gamma).
$$
Thus \b[the requested entropy-dissipation inequality is]
$$
\boxed{H(f_t\mid\gamma)\leq-\frac12\frac d{dt}H(f_t\mid\gamma).}
$$
It is the <Gaussian logarithmic Sobolev inequality> along this evolution. Since $H'\leq-2H$, an integrating factor gives \b[the <entropy convergence rate for Ornstein-Uhlenbeck flow>]:
$$
\boxed{H(f_t\mid\gamma)\leq e^{-2t}H(f_0\mid\gamma).}
$$
For finite initial entropy, $f_t$ therefore converges to the stationary <Gaussian density> in <relative entropy> at rate $2$. If desired, <Pinsker's inequality> also converts this to the density estimate $\|f_t-\gamma\|_{L^1}\leq\sqrt{2H(f_0\mid\gamma)}e^{-t}$.
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