English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We introduce and study the class of linear transfers between probabilitydistributions and the dual class of Kantorovich operators between functionspaces. Linear transfers can be seen as an extension of convex lowersemi-continuous energies on Wasserstein space, of cost minimizing masstransports, as well as many other couplings between probability measures towhich Monge-Kantorovich theory does not readily apply. Basic examples includebalayage of measures, martingale transports, optimal Skorokhod embeddings, andthe weak mass transports of Talagrand, Marton, Gozlan and others. The classalso includes various stochastic mass transports such as the Schr\"odingerbridge associated to a reversible Markov process, and the Arnold-Breniervariational principle for the incompressible Euler equations. We associate tomost linear transfers, a critical constant, a corresponding effective lineartransfer and additive eigenfunctions to their dual Kantorovich operators, thatextend Man\'e's critical value, Aubry-Mather invariant tori, and Fathi's weakKAM solutions for Hamiltonian systems. This amounts to studying the asymptoticproperties of the nonlinear Kantorovich operators as opposed to classicalergodic theory, which deals with linear Markov operators. This allows for theextension of Mather theory to other settings such as its stochasticcounterpart. We also introduce the class of convex transfers, which includes$p$-powers ($ p \geq 1$) of linear transfers, the logarithmic entropy, theDonsker-Varadhan information, optimal mean field plans, and certain freeenergies as functions of two probability measures, i.e., where the referencemeasure is also a variable. Duality formulae for general transfer inequalitiesfollow in a very natural way. This paper is an expanded version of a previouslyposted but not published work by the authors.

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