  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 \`a la Fathi-Mather, and ergodic
optimization of dynamical systems, we study the asymptotic properties of
general Kantorovich operators.
