An infinite-dimensional weak KAM theory via random variables

Diogo  Gomes; Levon  Nurbekyan

doi:10.3934/dcds.2016069

Article Contents

2016, Volume 36, Issue 11: 6167-6185. Doi: 10.3934/dcds.2016069

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An infinite-dimensional weak KAM theory via random variables

Diogo Gomes^1, and
Levon Nurbekyan^1,

1.
4700 King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900

Received: July 31, 2015

Revised: June 30, 2016

Published: May 2017

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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 $\mathbb{R}$.

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Mathematics Subject Classification: Primary: 37K05, 37K55; Secondary: 49L20, 49L25.

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