turkmath.org

The notion of a non-linear Kantorovich operator was motivated by the celebrated duality in the mass transport problem, hence the name. In retrospect, we realized that they -and their iterates- were omnipresent in several branches of analysis, even those that are focused on linear Markov operators and their semi-groups such as classical ergodic theory, potential theory, and probability theory. The Kantorovich operators that appear in these cases, though non-linear, are all positively 1-homogenous rendering most classical operations on measures and functions conducted in these theories cost-free. General Kantorovich operators arise when one assigns a cost to such operations. Kantorovich operators are also Choquet capacities and are the least non-linear extensions of Markov operators, which make them a relatively manageable subclass of non-linear maps. Motivated by the stochastic counterpart of Aubry-Mather theory for Lagrangian systems and Fathi-Mather weak KAM theory, as well as ergodic optimization of dynamical systems, we exhibit the asymptotic properties of general Kantorovich operators.