On optimal stationary couplings between stationary processes

By a classical result of [10] the ρ{variant} 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 ρ{variant} distance in these cases in explicit form. We also extend the ρ{variant} distance to random fields and to general nonmetric distance functions and give a construction method for optimal stationary c-couplings. Our assumptions need in this case a geometric positive curvature condition.