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Optimal Markovian couplings and applications. (English) Zbl 0813.60068
A measure on the product space is called a coupling of its two marginal measures. For a given distance function on the state space, the optimal coupling is the special one, under which the integral of the distance function w.r.t. this coupling measure takes minimum. Under some conditions on the distance function, the optimal coupling of birth-death processes and the optimal Markov coupling of birth-death chains and diffusion processes are given. The result is applied to obtain a nice lower bound of the spectral gap of the Laplace-Beltrami operator on a Riemannian manifold $$(M,g)$$ of Ricci curvature not less than $$-Kg$$ with nonnegative $$K$$.

##### MSC:
 60J27 Continuous-time Markov processes on discrete state spaces 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60J60 Diffusion processes
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