The mather measure and a large deviation principle for the entropy penalized method

Diogo Gomes*, A. O. Lopes, J. Mohr

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

We present the rate function and a large deviation principle for the entropy penalized Mather problem when the Lagrangian is generic (it is known that in this case the Mather measure μ is unique and the support of μ is the Aubry set). We assume the Lagrangian L(x, v), with x in the torus T N and v∈RN, satisfies certain natural hypotheses, such as superlinearity and convexity in v, as well as some technical estimates. Consider, for each value of and h, the entropy penalized Mather problem min TN ×RN L(x,v)dμ(x,v)+S[μ] where the entropy S is given by S[μ] = fTN×RNμ(x, v)lnμ(x, v)/fRNμ(x, w)dw, and the minimization is performed over the space of probability densities μ(x, v) on N×N that satisfy the discrete holonomy constraint ∫N× N φ(x + hv) - φ(x) dμ = 0. It is known [17] that there exists a unique minimizing measure μ, h which converges to a Mather measure μ, as , h→0. In the case in which the Mather measure μ is unique we prove a Large Deviation Principle for the limit lim , h→0 ln μ, h(A), where A ⊂ N×N. In particular, we prove that the deviation function I can be written as I(x,v)= L(x,v)φ(x)(v)H0, where φ0 is the unique viscosity solution of the Hamilton Jacobi equation, H(x),x)=H. We also prove a large deviation principle for the limit → 0 with fixed h. Finally, in the last section, we study some dynamical properties of the discrete time AubryMather problem, and present a proof of the existence of a separating subaction.

Original languageEnglish (US)
Pages (from-to)235-268
Number of pages34
JournalCommunications in Contemporary Mathematics
Volume13
Issue number2
DOIs
StatePublished - Apr 1 2011

Keywords

  • Aubry-Mather measure
  • HamiltonJacobi equation
  • Large Deviation Principle
  • discrete AubryMather problem
  • entropy penalized Mahler problem
  • subaction
  • viscosity solution

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Fingerprint Dive into the research topics of 'The mather measure and a large deviation principle for the entropy penalized method'. Together they form a unique fingerprint.

Cite this