Kantorovich operators are non-linear extensions of Markov operators and are omnipresent in several branches of mathematical analysis. The asymptotic behaviour of their iterates plays an important role even in classical ergodic, potential and probability theories, which are normally concerned with linear Markovian operators, semi-groups, and resolvents. The Kantorovich operators that appear implicitly in these cases, though non-linear, are all positively 1-homogenous. General Kantorovich operators amount to assigning "a cost" to most operations on measures and functions normally conducted "for free" in these classical settings. Motivated by extensions of the Monge-Kantorovich duality in mass transport, the stochastic counterpart of Aubry-Mather theory for Lagrangian systems, weak KAM theory à la Fathi-Mather, and ergodic optimization of dynamical systems, we study the asymptotic properties of general Kantorovich operators.

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