Linear transfers, Kantorovich operators, and their ergodic properties

In this work, we introduce and study a class of convex functionals on pairs of probability measures, the linear transfers, which have a structure that com- monly arises in the dual formulations of many well-studied variational prob- lems. We show that examples of linear transfers include a large number of well-known transport problems, including the weak, stochastic, martingale, and cost-minimising transports. Further examples include the balayage of measures, and ergodic optimisation of expanding dynamical systems, among others. We also introduce an extension of the linear transfers, the convex transfers, and show that they include the relative entropy functional and p-powers (p ≥ 1) of linear transfers. We study the properties of linear and convex transfers and show that the inf-convolution operation preserves their structure. This allows dual formu- lations of transport-entropy and other related inequalities, to be computed in a systematic fashion. Motivated by connections of optimal transport to the theory of Aubry- Mather and weak KAM for Hamiltonian systems, we develop an analog in the setting of linear transfers. We prove the existence of an idempotent operator which maps into the set of weak KAM solutions, an idempotent linear transfer that plays the role of the Peierls barrier, and we identify analogous objects in this setting such as the Mather measures and the Aubry set. We apply this to the framework of ergodic optimisation in the holonomic case.Science, Faculty ofMathematics, Department ofGraduat