Notation:
- $T >0 $ is a positive real number and $d \in \mathbb{N}$ is a positive natural number;
- Given a complete metric space $X$ I denote with $\mathcal{P}(X)$ (resp. $\mathcal{M}(X)$) the space of Borel probabilities (resp. finite positive measures) on $X$. $C_b(X)$ is the space of real continuous bounded functions on $X$;
- I denote with $\mathcal{L}^1$ the Lebesgue measure on the real line;
- Given a family $\{\mu_t\}_{t \in [0,T]} \subset \mathcal{P}(X)$ and a measure $\nu \in \mathcal{M}([0,T])$ I denote with $\mu_t \otimes \nu$ the measure $$\int_{X \times [0,T]} \varphi(x,t) \text{ d}(\mu_t \otimes \nu)(x,t) = \int_0^T \int_X \varphi(x,t) \text{ d} \mu_t(x) \text{ d} \nu(t) $$ for every $\varphi \in C_b(X \times [0,T])$.
Setting:
- $X= \mathbb{R}^d$;
- $\nu_k : [0,T] \to \mathcal{P}(X)$, $\sigma \in \mathcal{P}(X)$, $\mu_k \in \mathcal{M}([0,T])$, $\gamma_k : [0,T] \to \mathcal{P}(X \times X)$ with $k \in \mathbb{N}$.
- $\gamma_k(t)$ is an optimal plan (w.r.t. squared Euclidean distance) between $\nu_k(t)$ and $\sigma$, for every $t \in [0,T]$ and $k \in \mathbb{N}$.
- $\nu_k(\cdot) \overset{k}{\to} \mu : [0,T] \to \mathcal{P}(X)$ in the sense of the $2$-Wasserstein distance on $\mathcal{P}(X)$ and uniformly w.r.t $t$.
- $ \mu_k \rightharpoonup \mathcal{L}^1 |_{[0,T]}$
- $\Theta_k := \gamma_k(t) \otimes \mu_k \rightharpoonup \Theta =: \Theta(t) \otimes \mathcal{L}^1 |_{[0,T]} \in \mathcal{M}(X \times X \times [0,T])$ with $\Theta(t) \in \mathcal{P}(X \times X)$ for every $t \in [0,T]$.
Question:
It is easy to show that $\Theta(t)$ is an admissible plan between $\mu(t)$ and $\sigma$ for every $t \in [0,T]$ i.e. it has $\mu(t)$ and $\sigma$ as marginals.
Can we prove that it is also optimal?