Langevin Diffusion Approximation to Same Marginal Schr”{o}dinger Bridge

arXiv:2505.07647v2 Announce Type: replace-cross
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.