An infinite-dimensional Weak KAM theory via random variables

TL;DR

This paper extends Weak KAM theory to infinite-dimensional spaces using a random variables approach, establishing the existence of viscosity solutions, invariant measures, and calibrated curves.

Contribution

It introduces a novel random variables framework to develop infinite-dimensional Weak KAM theory, proving key existence results.

Findings

Existence of viscosity solutions for the infinite-dimensional cell problem

Presence of invariant minimizing measures in the infinite-dimensional setting

Existence of calibrated curves defined on the real line

Abstract

We develop several aspects of the infinite-dimensional Weak KAM theory using a random variables' approach. We prove that the infinite-dimensional cell problem admits a viscosity solution that is a fixed point of the Lax-Oleinik semigroup. Furthermore, we show the existence of invariant minimizing measures and calibrated curves defined on R.