Existence of the optimal measurable coupling and ergodicity for Markov processes
type: Item
instance of: scholarly article
title: Existence of the optimal measurable coupling and ergodicity for Markov processes
zbMATH Open document ID: 0938.60011
DOI: 10.1007/BF02872050
author: Shaoyi Zhang
published in: Science in China. Series A
publication date: 2000-01-05T00:00:00Z
review text: 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 \S 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.
Mathematics Subject Classification ID: 60B05
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 1274857
zbMATH Keywords: coupling
zbMATH Keywords: measurability
zbMATH Keywords: Dobrushin-Shlosman uniqueness theorem
reviewed by: Mu-Fa Chen
MaRDI profile type: Publication
cites work: Q3687243
cites work: Q4360710
cites work: Optimal markovian couplings and applications
cites work: Q4003728
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description: scientific article; zbMATH DE number 1274857