  We study the convexity of mutual information as a function of time along the
flow of the Ornstein-Uhlenbeck process. We prove that if the initial
distribution is strongly log-concave, then mutual information is eventually
convex, i.e., convex for all large time. In particular, if the initial
distribution is sufficiently strongly log-concave compared to the target
Gaussian measure, then mutual information is always a convex function of time.
We also prove that if the initial distribution is either bounded or has finite
fourth moment and Fisher information, then mutual information is eventually
convex. Finally, we provide counterexamples to show that mutual information can
be nonconvex at small time.
