Define a free energy functional on the space of probability densities (on $\mathbb{R}^d$, denoted $\mathcal{P}(\mathbb{R}^n)$)
$$E(\rho):=\int_{\mathbb{R}^d} f(x) \rho(x) dx+\int_{\mathbb{R}^d} \rho(x) \log \rho(x) dx $$ for some uniformly convex, Lipschitz, non-negative $f: \mathbb{R}^n\to \mathbb{R}$.
Consider the following discrete scheme of a Wasserstein gradient flow (coined JKO scheme) : Fix a time step $h$, given a density with finite second moment $\rho^0$ such that $E(\rho^0)<\infty$ iteratively define
$$\rho^{n+1}:=\text{argmin}_{\rho \in \mathcal{P}(\mathbb{R}^n)} \frac{1}{2h}W_2^2(\rho^n,\rho)+E(\rho).$$
Now for fixed $h$ consider the sequence $\{\rho^n\}_{n\ge 0}$.
My question: does anyone know of any results relating to the convergence of $\rho^n$ as $n \to \infty$ or, more likely, the existence of an accumulation point of $\{\rho^n\}_{n\ge 0}$?
Note - we are fixing the time step hence this is different question than that originally answered in this famous paper. I have been looking for results on the stability of the scheme and I can't find any (but im sure there must be).