I was wondering whether one has studied the Bregman divergence arising from a squared Wasserstein distance.
More precisely, let $\Omega\subset \mathbb{R}^d$ be a compact set and $c\in \Omega\times \Omega $ be continuous. Let $\nu\in P(\Omega)$ be a fixed probability measure, and define the map $P(\Omega) \ni \mu\mapsto T(\mu,\nu)\in \mathbb {R}$ such that $$ T(\mu,\nu)=\min_{\gamma \in \Pi(\mu,\nu)}\int_\Omega c(x,y)d \gamma, $$ where $\Pi(\mu,\nu)$ is the set of probability measures with $\mu$ and $\nu$ as their first and second marginals, respectively.
As shown in Proposition 7.17 of the book, $\mu\mapsto T(\mu,\nu)$ is convex and under suitable conditions on $\nu$ and $c$, for all $\mu,\mu'\in P(\Omega)$, $$ \lim_{\epsilon \to 0} \frac{T(\mu+\epsilon (\mu'-\mu),\nu)}{\epsilon}=\int_\Omega \varphi_\mu (x) (\mu'-\mu)(dx), $$ where $\varphi_\mu$ is the unique Kantorovich potential from $\mu$ to $\nu$. This allows us to define the following Bregman divergence $D(\cdot|\cdot):P(\Omega)\times P(\Omega)\to \mathbb{R}$ by $$ D(\mu'|\mu)=T(\mu',\nu)-T(\mu,\nu)-\int_\Omega \varphi_\mu (x) (\mu'-\mu)(dx), $$ where $\varphi_\mu (x)$ is the first variation of $ T(\cdot,\nu)$ at $\mu$.
Has the above Bregman divergence been studied in the literature? For instance, with specific choices of $c$, say $c(x,y)=|x-y|^2$, can we simplify the expression of $D(\mu'|\mu)$? Can we obtain its relation with commonly used metrics of probability measures?
