Disclaimer: theoretical physicists here! I apologize in advance for some sloppy terminology.
I recently found an interest in optimal transport theory for some potential application to my research field in physics and I have been trying to study some basics from the standard reviews by Villani and Santambrogio.
My main questions concern with the dynamical formulation of OT, especially with the fluid mechanics version by Benamou–Brenier $$ \min_{(v,\mu)} \left\{\int_0^1 \int_{\mathcal{M}} |v_t|^2 \text{d}\mu_t, \,\,\, \frac{\partial \mu_t}{\partial t} + \nabla\cdot(\mu_t v_t) = 0\right\}. $$ From here forward let us assume we are working with a quadratic cost and the underlying space is some Riemannian manifold $\mathcal{M}$. Here are some of my questions:
- When discussing the Benamou–Brenier formulation, as I understand it, we are essentially trying to solve for the velocity field $v_t$ and the one-parameter family of measures $\mu_t$ that minimize the total kinetic energy subject to a constraint given by the continuity equation. Furthermore, we impose some boundary conditions on $\mu_0$ and $\mu_1$. The way I understand things is that the optimal transport map $T^t(x)$ is given by the exponential map at $x$ of some convex function $\phi$. I also know that the transport map also induces the curve $\mu_t$ that lives on the space of measures $\mathcal{P}(\mathcal{M})$ with Wasserstein metric. My question then is in regards of the velocity field $v_t$: should I think of it as being the tangent vector to the geodesic on $\mathcal{M}$, or is it a more abstract vector field living on $\mathcal{P}(\mathcal{M})$ and tangent to the curve of $\mu_t$? How should I interpret the continuity equation, then?
In regards of the Kantorovich potential $\phi$ that generates the optimal transport map -- how is this object found exactly? It should obey an Hamilton-Jacobi equation of some sort, correct?
Finally, in the formulation of the problem, instead of setting $\mu_0$ and $\mu_1$ as given, could we replace that with some Cauchy initial conditions given by $\mu_0$ and the initial vector $v_0$?
Hope the questions are clear, and again, I apologize for some potential sloppiness!