New answers tagged

1 vote
Accepted

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 ...
leo monsaingeon's user avatar
0 votes

Gradient flows: evolution of geodesics

As currently asked the answer is NO, because your desired upper bound already fails for $t=0$ (or equivalently, $t=1$). Indeed, it is well understood that the small-time deviation along the heat flow, ...
leo monsaingeon's user avatar
9 votes

Proving the inequality involving Hausdorff distance and Wasserstein infinity distance

EDIT: answer 2 below is completely false, as pointed out by the OP. However this is such a typical example of wishful thinking that I believe it is worth leaving for the posterity. (I'll record it ...
leo monsaingeon's user avatar

Top 50 recent answers are included