Initialization Stability and Convergence of Projected Langevin Dynamics

The entropic regularized optimal transport (EOT) plays a fundamental role in the computation of optimal transport. In \cite{CLP}, the authors proposed a gradient flow approach to EOT, and derived a projected Langevin diffusion that converges in distribution to the solution of EOT when initialized with a coupling between the two given marginals. In this paper, motivated by samplings of optimal couplings for EOT, we analyze the projected Langevin dynamics with arbitrary initial distribution, and prove that its long time behavior is the same as if it was initialized from a coupling.