I am exploring an idea which is based on computing the Wasserstein (Kantorovich) distance between the empirical transition probability densities of two Itô diffusion processes.
Following from this paper by A. Milian, the transition probability of the process $X(t)$
$$P(t,x,s,y) = Pr \left[ X(s) \le y | X(t) = x \right], t<s$$
has the density function $p(t,x,s,y)$.
In this problem I am working on I consider a pair of processes, $X(t)$ and $Y(t)$. Let the transition density function of $X(t)$ be $p(t,x,s,y)$ and of $Y(t)$ be $q(t,x,s,y)$. And I wish to write out an expression for the Wasserstein (Kantorvich) distance between $p(t,x,s,y)$ and $q(t,x,s,y)$.
First we should define the Wasserstein-p distance between two probability measures $\mu$ and $\nu$ on a metric space $M$ as
$$W_p (\mu, \nu)= \left( \inf_{\gamma \in \Gamma (\mu, \nu)} \mathbb{E}_{ (x,y)\sim \gamma } d(x, y)^p \right)^{1/p}$$
where $\Gamma(\mu, \nu)$ is the set of couplings of $\mu$ and $\nu$. The gap that I am trying to bridge here, and write out correctly is that we consider $p(t,x,s,y)$ and $q(t,x,s,y)$ to define two probability measures on the same metric space.
I have been attracted to the dual form of the Wasserstein distance, which for $p$ and $q$ above I would write like this. Using the $p=1$ form of Wasserstein distance in fine for my application, I believe.
$$W_1(p,q)= \sup_{f \in \mathcal{F}} \left[ \int f(x) d p(x) - \int f(x) d q(x) \right]$$
Where $\mathcal{F}$ is the family of all continuous functions with Lipschitz constant less than or equal to 1. For any function $f \in \mathcal{F}$, then this previous equation is equivalent to writing the following.
$$\mathbb{E}_{x \sim p} [f(x)] - \mathbb{E}_{y \sim q} [f(y)] \le W_1(p,q)$$
This is pretty much as far as I have got. Is the way I've written this allowed, OK? What I am not super satisfied with is the transition from the expression of the transition density which is a function of 4 variables to the more general form of a probability measure that is contained in the definitions for the Wasserstein distance.
Is there a way to write the Wasserstein distance between $p$ and $q$ which includes all 4 space/time variables, $t, x, s, y$?