Transport information Bregman divergences

We study Bregman divergences in probability density space embedded with the $L^2$--Wasserstein metric. Several properties and dualities of transport Bregman divergences are provided. In particular, we derive the transport Kullback--Leibler (KL) divergence by a Bregman divergence of negative Boltzmann--Shannon entropy in $L^2$--Wasserstein space. We also derive analytical formulas and generalizations of transport KL divergence for one-dimensional probability densities and Gaussian families.