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Weak KAM Theory topics in the stationary ergodic setting

  • Andrea Davini
  • Antonio SiconolfiEmail author
Article

Abstract

We perform a qualitative analysis of the critical equation associated with a stationary ergodic Hamiltonian through a stochastic version of the metric method, where the notion of closed random stationary set, issued from stochastic geometry, plays a major role. Our purpose is to give an appropriate notion of random Aubry set, to single out characterizing conditions for the existence of exact or approximate correctors, and write down representation formulae for them. For the last task, we make use of a Lax-type formula, adapted to the stochastic environment. This material can be regarded as a first step of a long-term project to develop a random analog of Weak KAM Theory, generalizing what done in the periodic case or, more generally, when the underlying space is a compact manifold.

Mathematics Subject Classification (2000)

35D40 35B27 35F21 49L25 

List of symbols

N

Integer number

BR(x0)

Closed ball in \({\mathbb R^N}\) centered at x 0 of radius R

BR

Closed ball in \({\mathbb R^k}\) centered at 0 of radius R

\({\langle\,\cdot\;, \cdot\,\rangle}\)

Scalar product in \({\mathbb R^N}\)

| · |

Euclidean norm in \({\mathbb R^N}\)

\({\mathbb R_+}\)

Set of nonnegative real numbers

\({\mathcal{B}(\mathbb R^k)}\)

σ-Algebra of Borel subsets of \({\mathbb R^k}\)

χE

Characteristic function of the set E

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Dip. di MatematicaUniversità di Roma “La Sapienza”RomaItaly

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