  We take a new look at the relation between the optimal transport problem and
the Schr\"{o}dinger bridge problem from the stochastic control perspective. We
show that the connections are richer and deeper than described in existing
literature. In particular: a) We give an elementary derivation of the
Benamou-Brenier fluid dynamics version of the optimal transport problem; b) We
provide a new fluid dynamics version of the Schr\"{o}dinger bridge problem; c)
We observe that the latter provides an important connection with optimal
transport without zero noise limits; d) We propose and solve a fluid dynamic
version of optimal transport with prior; e) We can then view optimal transport
with prior as the zero noise limit of Schr\"{o}dinger bridges when the prior is
any Markovian evolution. In particular, we work out the Gaussian case. A
numerical example of the latter convergence involving Brownian particles is
also provided.
