Langevin Diffusion Approximation to Same Marginal Schr"odinger Bridge

We introduce a novel approximation to the same marginal Schr"odinger 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"odinger 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 L2. 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"odinger 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.