When the m marginal constraints are the time-marginals of an SDE with time-dependent drift, the multi-marginal Schrödinger bridge converges to the SDE's law at KL rate O(m^{-1}).
Stochastic control liaisons: Richard sinkhorn meets gaspard monge on a schr¨ odinger bridge.Siam Review, 63(2):249–313, 2021
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Convergence Rate of the Solution of Multi-marginal Schrodinger Bridge Problem with Marginal Constraints from SDEs
When the m marginal constraints are the time-marginals of an SDE with time-dependent drift, the multi-marginal Schrödinger bridge converges to the SDE's law at KL rate O(m^{-1}).