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Existence of the optimal measurable coupling and ergodicity for Markov processes. (English) Zbl 0938.60011
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 \(E\). The author proves that for given two transition probabilities \(P_1(x_1, d y_1)\) and \(P_2(x_2, d y_2)\), there always exists a coupled transition probability \(P(x_1, x_2, dy_1, dy_2)\) such that \[ \int \rho (y_1, y_2) P(x_1, x_2, dy_1, dy_2) = W(P_1(x_1, \cdot), P_2(x_2, \cdot)) \] for all \(x_1, x_2\in E\), where \(W(P_1, P_2)\) is the Wasserstein distance of probability measures \(P_1\) 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 [Acta Math. Sin., Engl. Ed.], the author extends the above result to nonnegative, lower semi-continuous function instead of distance \(\rho\). This enables the author to solve an open problem about stochastic comparison problem. 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.

MSC:
60B05 Probability measures on topological spaces
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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[1] Dobrushin, R. L., Shlosman, S. B Constructive criterion for the uniqueness of Gibbs field, inStatistical Mechanics Dynamical Systems (ed. Frite, J.), Boston: Birkhauser, 1985, 347–370. · Zbl 0569.46042
[2] Zhang, S. Y., Xu Kau, On the existence of the optimal measurable coupling of transition probability,Acta Math. Sin. (in Chinese), 1997, 40(1): 5. · Zbl 0884.60088
[3] Chen, M. F., Optimal Markovian couplings and applications,Acta Math. Sin., New. Ser., 1994, 10(3): 260. · Zbl 0813.60068
[4] Chen, M. F.,From Markov Chains to Non-equilibrium Particle Systems, Singapore: World Scientific, 1992. · Zbl 0753.60055
[5] Zhang, S. Y., The existence of the {\(\rho\)}-optimal coupling operator for jump process,ActaMath. Sin. (in Chinese), 1998, 41 (2): 397.
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