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Let $\mu : [0, T] \to \mathcal P_2^a (\mathbb R^d), t \mapsto \mu_t$ be absolutely continuous. Is $t \mapsto \mathcal H (\mu_t)$ continuous?
$\newcommand{\R}{\mathbb R}$The answer is NO. I will provide below a counterexample in dimension $d=1$.
Preliminaries:
Let's agree that the entropy is
$$
H(\rho)=\int_{\mathbb R}\rho(x)\log\rho(x) dx
...
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