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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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##### References:
  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  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  Chen, M. F., Optimal Markovian couplings and applications,Acta Math. Sin., New. Ser., 1994, 10(3): 260. · Zbl 0813.60068  Chen, M. F.,From Markov Chains to Non-equilibrium Particle Systems, Singapore: World Scientific, 1992. · Zbl 0753.60055  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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