I have time dependant measures, say $\mu: [0,T] \rightarrow \mathcal{M}(\Omega)$ and I'm looking to define a penalization in this space that would measure the amount of displacement in space through time.
My first option is to consider the Wasserstein metric. The simplest example I can think of is that for two time measurements $0\leq t_1<t_2 \leq T$, the following penalization can be considered $$ P_2(\mu) = W_2(\mu(t_1), \mu(t_2))$$
More generally, I could consider for $0 = t_0 < \ldots < t_n = T$ $$ P_n(\mu) = \frac{1}{N} \sum_{i=0}^{n-1} W_2(\mu(t_i), \mu(t_{i+1}))$$
So my question is, what happens when the limit $\lim_{n\rightarrow \infty} P_n(\mu)$ is considered? Does it has any name I could search in the literature?