Relative Fisher information decay under Ornstein-Uhlenbeck flow ($I(t)\le e^{-2(t-s)}I(s)$)

The differential estimate $I^{\prime }(t)\le -2I(t)$ gives the displayed exponential bound by an integrating factor. Start at any time with finite information. Combining this with a zero limiting entropy yields $H(f_t\mid \gamma )={\int }_t^{\infty }I(s)ds\le I(t)/2$.