Exponential Convergence of the Sinkhorn Algorithm for the Schrödinger Bridge with Regime Switching

This paper studies the convergence of the Sinkhorn algorithm for the Schr\"odinger bridge problem with regime switching, as introduced in Zlotchevski and Chen (2025). We consider a class of regime-switching stochastic systems on the hybrid state space $E=\mathbb{R}^d\times\{1,\ldots,m\}$, and construct the Sinkhorn iteration through the associated equivalent entropic optimal transport formulation. The main result of this paper is the exponential convergence of the Sinkhorn algorithm in relative entropy under compactness assumptions. Our proofs are inspired by the arguments recently developed for proving exponential convergence of the classical Schr\"odinger problem in Chiarini, Conforti, Greco and Tamanini (2024), Eckstein (2025). We perform a similar analysis for the partially observed terminal setting, where only the marginal distribution of the continuous component is prescribed at the terminal time, while the discrete regime is unobserved.