Monotonicity, Convexity, and Smoothness of the KL-Divergence between Two Brownian Motions with Different Initializers We consider the two distributions
$$
p_t = p_0 * N(0, tI),\quad q_t  = q_0 * N(0, t I),
$$
where $*$ denotes the convolution between two densities, while $p_0$ and $q_0$ have the same mean and variance. In particular, we assume that $q_0$ is $N(0, I)$. In other words, we consider two random variables
$$
X_t = X_0 + N(0, t I)\sim p_t, \quad Y_t = Y_0 + N(0, tI)\sim q_t,
$$
where $X_0\sim p_0$ and $Y_0\sim q_0 = N(0, I)$. 
We are interested in characterizing the evolution of 
$$
\text{KL}(p_t, q_t) = \int p_t \log \left(\frac{p_t}{q_t}\right) dx
$$
along $t$. In particular, we aim to prove:
(i) $\text{KL}(p_t, q_t)$ is monotonically decreasing along $t$;
(ii) $\text{KL}(p_t, q_t)$ is convex along t;
(iii) $\text{KL}(p_t, q_t)$ is smooth along t, that is, 
$\frac{d^2\text{KL}(p_t, q_t)}{dt^2}
$ is upper bounded. 
(Comment: As pointed out by Jon, (i) follows directly from the data processing inequality. As pointed out by Nawaf, (ii) also holds.)
 A: Write the KL divergence in terms of the differential entropy of the random variables $X_t$ and $Y_t$; the result quickly follows.  Indeed, since $Y_t \sim \mathcal{N}(0,1+t)$, we have \begin{align*}
\operatorname{KL}(p_t, q_t) &= - h(p_t) + \frac{1}{2} \int \frac{x^2}{1+t} p_t(x) dx + \frac{1}{2} \log(1+t) + \frac{1}{2} \log(2 \pi) \\
&= - h(p_t) + \frac{1}{2} \mathbb{E}((X_0+\sqrt{t} \mathcal{N}(0,1))^2)\frac{1}{1+t}  + \frac{1}{2} \log(1+t) + \frac{1}{2} \log(2 \pi) \\
&= -h(p_t) + h(q_t)
\end{align*} where $h(\cdot)$ is the differential entropy.  By Lemma 2 of Zhang, Anantharam and Geng, subject to $\operatorname{var}(X_0)=1$, the minimum of $-\frac{d^2}{dt^2} h(p_t)$ is achieved when $X_0$ is Gaussian.  Thus, $-\frac{d^2}{dt^2} h(p_t) \ge -\frac{d^2}{dt^2} h(q_t)$, and hence, $\frac{d^2}{dt^2}\operatorname{KL}(p_t, q_t) \ge 0$ which implies that $\operatorname{KL}(p_t, q_t)$ is convex with respect to $t$.
ADD
The Gaussian minimality result used above seems to go back to
McKean, H. P., Speed of approach to equilibrium for Kac’s caricature of a Maxwellian gas, Arch. Ration. Mech. Anal. 21, 343-367 (1966). ZBL1302.60049.
