Equivalent definition of the Kantorovich-Fisher-Rao distance

Question

I am reading this paper

"A JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows" (https://arxiv.org/abs/1602.04457)

and in the proof of Proposition 2.2, basically, if the measure $\rho$ is smooth such that $\rho = \rho(x)dx$, i.e., we think about the measure as its density then one can define

$$\mathbf{v}^\varepsilon = \mathbf{v} + \varepsilon \frac{\mathbf{w}}{\rho}$$

where $\mathbf{v}$ is a given vector field and $\mathbf{w}$ is any divergence-free vector field, and the proof carries through.

My question is: if $\rho$ is not that nice, how to define such a thing? I guess some kind of approximations but not really know what to do?

Answer 1

Well, when we wrote the paper we were not really concerned with full rigor at this stage, all we wanted to emphasize was that the "KFR" distance (by now rather the WFR or HK distance, as in Wasserstein-Fisher-Rao or Hellinger-Kantorovich) constructed in 3 independent papers was really the same, at least formally. To the best of my knowledge this still hasn't been proved fully rigorously, but it is clear to everybody in the field (yes, I know how bad this may sound...)

Regarding your precise question: if you wish, our proposition 2.2. is a purely formal result, so the proof is formal too. It is not one of these proofs where the statement is fully rigorous, you first prove it under stronger (regularity/positivity) assumptions, and then relax the assumptions to get full generality. Our statement is formal to begin with (and I would agree that we should have mentioned this very explicitly when writing the paper), hence so is the proof.

For a partial attempt at answering more mathematically: even in classical optimal transport, the statement that the velocity field $v_t=\nabla r_t$ can be taken as a gradient is to be understood in a very delicate and particular sense. (Our whoe point in the paper was that the $r_t$ in question is the reaction term, so horizontal displacement and vertical creation/anihilation of mass are related.) One of the reasons for this is that, if one thinks of an (optimal, kinetic-energy-minimizing) velocity field $v_t$ as representing a "tangent vector" $T_{\mu_t}\mathcal P$ at a point $\mu_t$ in the Wasserstein space, then the natural functional space to have it live in is the weighted $L^2(\mu)t)$ space. As a consequence, that "$v_t$ must be a gradient" should rather be replaced by the condition that $$ v_t\in \overline{\{\nabla\phi,\,\phi\in C^1\}}^{L^2(\mu_t)}, $$ where the completion of smooth gradients is taken in the above and natural functional setting. This is exactly how the "tangent space" is constructed for optimal transport, I can suggest for example to look into [1], sections 8.4 and 8.5. Taking the completion precisely takes care of the "vacuum issue" (that $\rho$ may not be smooth and positive).

[1] Ambrosio, L., Gigli, N., & Savaré, G. (2005). Gradient flows: in metric spaces and in the space of probability measures. Springer Science & Business Media.