Nonlocal Onsager Operators and Entropy Dissipation for Finite-State Schr\"odinger Bridges

Abstract

We investigate the Schr\"odinger bridge problem on a finite state space with a strictly positive Markov reference kernel. Starting from the semi-dual convex formulation, we introduce a continuous-time evolution for the terminal Schr\"odinger potential and show that its equilibria coincide with the unique solution of the bridge problem. The proposed dynamics induces an evolution for the terminal marginal. This marginal equation is governed by a state-dependent nonlocal Onsager operator, identified with the Hessian of the semi-dual functional. We derive its associated Dirichlet form, establish coercivity estimates on the appropriate quotient space, and interpret the resulting equation as a nonlocal gradient-flow formulation of relative entropy. Under natural positivity assumptions, we prove global well-posedness of the SBOF, convergence to the Schr\"odinger bridge, convergence of the induced couplings and path measures, and exponential relaxation of the terminal marginal. The latter follows from a uniform Poincar\'e inequality on compact sublevel sets together with entropy--variance comparison estimates. We also discuss the connection with finite-state generative modeling through the Doob transform and illustrate the theory on finite-grid examples involving rare states.