Minimax probabilities for Aubry-Mather Problems

TL;DR

This paper investigates minimax Aubry-Mather measures in discrete and continuous time, establishing existence, non-existence, and characterizations, with examples illustrating the differences between the two settings.

Contribution

It provides a comprehensive analysis of minimax Aubry-Mather measures, including existence results, duality properties, and a complete characterization of periodic measures in continuous time.

Findings

Existence of minimax measures in discrete time

Non-existence of stationary minimax measures in continuous time

Complete characterization of periodic minimax measures in continuous time

Abstract

In this paper we study minimax Aubry-Mather measures and its main properties. We consider first the discrete time problem and then the continuous time case. In the discrete time problem we establish existence, study some of the main properties using duality theory and present some examples. In the continuous time case, we establish both existence and non-existence results. First we give some examples that show that in continuous time stationary minimax Mather measures are either trivial or fail to exist. A more natural definition in continuous time are $T$-periodic minimax Mather measures. We give a complete characterization of these measures and discuss several examples.