Abstract
We introduce a novel approximation to the same marginal Schr\"{o}dinger bridge using the Langevin diffusion. As \varepsilon \downarrow 0, it is known that the barycentric projection (also known as the entropic Brenier map) of the Schr\"{o}dinger bridge converges to the Brenier map, which is the identity. Our diffusion approximation is leveraged to show that, under suitable assumptions, the difference between the two is \varepsilon times the gradient of the marginal log density (i.e., the score function), in \mathbf{L}^2. More generally, we show that the family of Markov operators, indexed by \varepsilon > 0, derived from integrating test functions against the conditional density of the static Schr\"{o}dinger bridge at temperature \varepsilon, admits a derivative at \varepsilon=0 given by the generator of the Langevin semigroup. Hence, these operators satisfy an approximate semigroup property at low temperatures.
