Mass splitting under adapted Wasserstein distance

In classical optimal transport, simply by Jensen’s inequality we have

\[\mathcal{W}_1(\frac{1}{N}\sum_{n=1}^N \mu_n, \frac{1}{N}\sum_{n=1}^N \nu_n) \leq \frac{1}{N}\sum_{n=1}^N \mathcal{W}_1(\mu_n, \nu_n).\]

However in adapted optimal transport, this is in general not true. A simple counter example is by taking \(\mu_1 = \delta_{(0,1)}\), \(\mu_2 = \delta_{(0,-1)}\), \(\nu_1 = \delta_{(\varepsilon,1)}\), \(\nu_2 = \delta_{(-\varepsilon,-1)}\). Then we have

\[\mathcal{AW}_1(\frac{1}{N}\sum_{n=1}^N \mu_n, \frac{1}{N}\sum_{n=1}^N \nu_n) = 1-\varepsilon > \varepsilon = \frac{1}{N}\sum_{n=1}^N \mathcal{AW}_1(\mu_n, \nu_n).\]

Conjecture: under some regularity conditions (quantified by \(\epsilon\)), we have

\[\mathcal{AW}_1(\frac{1}{N}\sum_{n=1}^N \mu_n, \frac{1}{N}\sum_{n=1}^N \nu_n) \leq \frac{1}{N}\sum_{n=1}^N \mathcal{AW}_1(\mu_n, \nu_n) + C_\epsilon.\]