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On the existence of the optimal measurable coupling of transition probability. (Chinese. English summary) Zbl 0884.60088
This paper studies Markovian coupling for a given transition function \(P(x, dy)\) on a Polish space \((E, \rho, \mathcal E)\), where \(\rho\) is a metric on \(E\). Roughly speaking, the author proves that if the family \(\mathcal M:=\{P(x, \cdot): x\in E\}\) is tight, then there exists a coupled transition probability \(P(x_1, x_2, dy_1, dy_2)\) such that \[ \int \rho (y_1, y_2) P(x_1, x_2, dy_1, dy_2) = W(P(x_1, \cdot), P(x_2, \cdot)) \] for all \(x_1, x_2\in E\), where \(W(P_1, P_2)\) is the Wasserstein distance of \(P_1\) and \(P_2\). Originally, the problem comes from the well-known Dobrushin-Shlosman uniqueness theorem for random fields. In the original proof, the measurability of \(P(x_1,x_2, dy_1, dy_2)\) in \((x_1, x_2)\) was missed. See also the reviewer’s book “From Markov chains to non-equilibrium particle systems” (1992; Zbl 0753.60055), Theorem 10.9 and §10.8. Very recently, in a forthcoming paper, the authors improved the above result by removing the tightness assumption. Thus, the authors finally established the existence theorem of \(\rho\)-optimal Markovian coupling for time-discrete Markov processes. Refer to the reviewer’s paper [Acta Math. Sin., New Ser. 10, No. 3, 260-275 (1994; Zbl 0813.60068)] for further background of the study on optimal couplings. The authors have also extended the above result to the time-continuous jump processes.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60H05 Stochastic integrals