Mather Measures and Ergodic Properties of Kantorovich Operators associated to General Mass Transfers

We introduce and study the class of linear transfers between probability distributions and the dual class of Kantorovich operators between function spaces. Linear transfers can be seen as an extension of convex lower semi-continuous energies on Wasserstein space, of cost minimizing mass transports, as well as many other couplings between probability measures to which Monge-Kantorovich theory does not readily apply. Basic examples include balayage of measures, martingale transports, optimal Skorokhod embeddings, and the weak mass transports of Talagrand, Marton, Gozlan and others. The class also includes various stochastic mass transports such as the Schr\"odinger bridge associated to a reversible Markov process, and the Arnold-Brenier variational principle for the incompressible Euler equations. We associate to most linear transfers, a critical constant, a corresponding effective linear transfer and additive eigenfunctions to their dual Kantorovich operators, that extend Man\'e's critical value, Aubry-Mather invariant tori, and Fathi's weak KAM solutions for Hamiltonian systems. This amounts to studying the asymptotic properties of the nonlinear Kantorovich operators as opposed to classical ergodic theory, which deals with linear Markov operators. This allows for the extension of Mather theory to other settings such as its stochastic counterpart. We also introduce the class of convex transfers, which includes $p$-powers ($ p \geq 1$) of linear transfers, the logarithmic entropy, the Donsker-Varadhan information, optimal mean field plans, and certain free energies as functions of two probability measures, i.e., where the reference measure is also a variable. Duality formulae for general transfer inequalities follow in a very natural way. This paper is an expanded version of a previously posted but not published work by the authors.