I have recently been studying the theory of computational optimal transport and am very interested in how to describe the similarity between two optimal transport plans. Specifically, suppose $P_1,Q_1,P_2,Q_2$ are probability measures on $\Omega \subseteq \mathbf R^d$, we have two Monge problem:
$$
\min_{T\in L(P_1,Q_1)}\int \frac{1}{2}\|x-T(x)\|^2\,\mathrm{d}P_1(x)$$ $$\min_{T\in L(P_2,Q_2)}\int \frac{1}{2}\|x-T(x)\|^2\,\mathrm{d}P_2(x),
$$
where $L(P_1,Q_1)$,$L(P_2,Q_2)$ are sets of admissible maps. We can also consider their Kantorovitch relaxations, which are to calculate 2-Wasserstein distances
$$
\begin{split}
\frac{1}{2}W_2^2(P_1,Q_1)&=\min_{\pi \in \Pi(P_1,Q_1)}\int \frac{1}{2}\|x-y\|^2\,\mathrm{d}\pi(x,y)\\
\frac{1}{2}W_2^2(P_2,Q_2)&=\min_{\pi \in \Pi(P_2,Q_2)}\int \frac{1}{2}\|x-y\|^2\,\mathrm{d}\pi(x,y),
\end{split}
$$ where $\Pi(P_1,Q_1)$ and $\Pi(P_2,Q_2)$ are the sets of couplings. Under certain conditions, using Brenier's theorem, it is known that the solutions to Monge problems exist and they are corresponding to their optimal plans. Meanwhile, we also have dual formulations:
$$
\begin{split}
\frac{1}{2}W_2^2(P_1,Q_1) &=\sup_{(f,g) \in \Phi(P_1,Q_1)}\int f\,\mathrm{d}P+\int g\,\mathrm{d}Q\\
\frac{1}{2}W_2^2(P_2,Q_2) &=\sup_{(f,g) \in \Phi(P_2,Q_2)}\int f\,\mathrm{d}P+\int g\,\mathrm{d}Q,
\end{split}
$$ and supremes are achieved by pairs of functions(optimal potentials).
Question. Assuming that the 2-Wasserstein distances between
$P_1$ and $P_2$, $Q_1$ and $Q_2$ are both very small, can it be inferred that the two optimal transport plans are "similar"?
There are some ambiguities here that I'd like to clarify: first, what it means for optimal transport to be "similar" is something I'm also curious about. My thought is whether we can use the optimal potentials to describe the similarity, meaning, would the optimal potentials of the two optimal transport problems be close to each other as functions (for example, having a small L1 distance between them)? Another perspective is whether the KL divergence between the optimal plans would be relatively small? I realize my question is not clearly articulated, so here I'm just looking to find out if there is any theoretical research on this matter and would appreciate any literature or links related to this background.