  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.
