I am considering the minimizing movement scheme related to the gradient of entropy functional in 2-Wasserstein space. The problem is to minimize the following functional for each fixed $\eta$ which is a probability density w.r.t. $(\mathbb{R}^d,Leb)$ with finite second moments: $$\int\rho\log\rho 
 dx+W\_2^2(\rho,\eta),$$ among all probability densities $\rho$(so $\rho dx\ll Leb$) with finite second moments. Now I need to show the existence of a minimizer to this problem.


So first we choose a minimizing sequence $\rho\_n$, which gives that $W\_2^2(\rho\_n,\eta)$ are uniformly bounded. Since the second moments can be bounded by the 2-Wasserstein distance, we know the second moments of $\rho\_n$ are uniformly bounded, so they are tight(and also uniformly integrable). This gives a subsequence $\rho\_{n\_k}$ converging weakly to some probability measure $\mu$. Now we need to show $\mu\ll Leb$ and has finite second moment.


For the second part I used Skorokhod's theorem to find $X\_n\sim\rho\_n$ and $X\sim\mu$ with $X\_n\overset{a.s.}{\rightarrow}X$. Then Fatou's lemma gives $\mathbb{E}X^2\leq\liminf\_{n\rightarrow\infty}\mathbb{E}X\_n^2<\infty$.


But I have no idea how to show $\mu\ll Leb$: we can find counterexamples if we only have $X\_n$ converges a.s. and in $L^1$. We might need other observations; or it is possible that the limit of the minimizing sequence of this problem is not absolutely continuous w.r.t. Lebesgue measure?



How to prove the limit of minimizing sequence of measures is again absolutely continuous(w.r.t. Lebesgue) in the minimizing movement scheme?
$\newcommand{\ep}{\varepsilon}\newcommand\R{\mathbb R}$Yes, the minimizer $\mu$ is absolutely continuous (w.r. to the Lebesgue measure $|\cdot|$).


Indeed, you showed that
\begin{equation\*}
 F(\rho\_n)\to m:=\inf\_\rho F(\rho) \tag{-1}
\end{equation\*}
and
\begin{equation\*}
 \mu\_{\rho\_n}\to\mu \tag{-0.5}
\end{equation\*}
weakly for some sequence $(\rho\_n)$ of probability densities and some probability measure $\mu$, where
\begin{equation\*}
 F(\rho):=\int\rho\ln\rho\,dx+W\_2^2(\rho,\eta) \tag{0}
\end{equation\*}
and $\mu\_\rho(dx):=\rho(x)\,dx$.


Take any set $E\subseteq\R$ with $|E|=0$. We have to show that then $\mu(E)=0$.


Take any real $\ep>0$. By the regularity of the Lebesgue measure, there is an open set $G\_\ep\subset\R$ such that
\begin{equation\*}
\text{$E\subseteq G\_\ep$ and $|G\_\ep|<\ep$.} \tag{0.5} 
\end{equation\*}
By (-0.5) and the [Portmanteau theorem](https://en.wikipedia.org/wiki/Convergence_of_measures#Weak_convergence_of_measures),
\begin{equation\*}
 \mu(G\_\ep)\le\liminf\_n\mu\_{\rho\_n}(G\_\ep). \tag{1}
\end{equation\*}
Next, for each real $a>1$,
\begin{equation\*}
 \mu\_{\rho\_n}(G\_\ep)=K\_n+L\_n, \tag{2}
\end{equation\*}
where
\begin{equation\*}
 K\_n:=\int\_{G\_\ep\cap[\rho\_n\le a]}\rho\_n\,dx,\quad L\_n:=\int\_{G\_\ep\cap[\rho\_n>a]}\rho\_n\,dx,
\end{equation\*}
$[\rho\_n\le a]:=\rho\_n^{-1}((-\infty,a])$, $[\rho\_n>a]:=\rho\_n^{-1}((a,\infty))$.
Further,
\begin{equation\*}
 K\_n\le a|G\_\ep|<a\ep \tag{3}
\end{equation\*}
by (0.5), and
\begin{equation\*}
 L\_n\le \int\_{[\rho\_n>a]}\rho\_n\,dx
 \le\frac1{\ln a}\int \rho\_n\ln\rho\_n\,dx\le \frac{m+1}{\ln a} \tag{4}
\end{equation\*}
for all large enough $n$, by (-1) and (0).


By (0.5), (1), (2), (3), (4),
\begin{equation\*}
 \mu(E)\le\mu(G\_\ep)\le a\ep+\frac{m+1}{\ln a}, 
\end{equation\*}
for all real $\ep>0$ and all real $a>1$. Letting now $\ep\downarrow0$ and then $a\to\infty$, we get $\mu(E)=0$, as desired.


