On optimal stationary couplings between stationary processes

By a classical result of Gray, Neuhoff and Shields (1975) the $\bar\varrho$ distance between stationary processes is identified with an optimal stationary coupling problem of the corresponding stationary measures on the infinite product spaces. This is a modification of the optimal coupling problem from Monge--Kantorovich theory. In this paper we derive some general classes of examples of optimal stationary couplings which allow to calculate the $\bar\varrho$ distance in these cases in explicit form. We also extend the $\bar\varrho$ distance to random fields and to general nonmetric distance functions and give a construction method for optimal stationary $\bar c$-couplings. Our assumptions need in this case a geometric positive curvature condition.