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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

An infinite-dimensional weak KAM theory via random variables

Pages: 6167 - 6185, Volume 36, Issue 11, November 2016      doi:10.3934/dcds.2016069

 
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Diogo Gomes - 4700 King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia (email)
Levon Nurbekyan - 4700 King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia (email)

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 $\mathbb{R}$.

Keywords:  Dynamical systems, weak KAM theory, Hamilton-Jacobi equations, voscosity solutions.
Mathematics Subject Classification:  Primary: 37K05, 37K55; Secondary: 49L20, 49L25.

Received: August 2015;      Revised: July 2016;      Available Online: August 2016.

 References