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.)