Before revision:  Instead, we study a new problem, called the optimal transition coupling problem, in which the optimal transport problem is constrained to the set of stationary Markovian couplings satisfying a certain transition matrix condition. After drawing a connection between this problem and Markov decision processes, we prove that solutions can be obtained via the policy iterationalgorithm .
After revision:  Instead, we study a new problem, called the optimal transition coupling (OTC) problem correspond to alignments of the two chains that minimize long-term average cost. We establish a connection between this problem and Markov decision processes, we prove that solutions can be obtained via the policy iterationalgorithm .