{
  "id": "0707.2603",
  "version": "v2",
  "published": "2007-07-17T21:28:34.000Z",
  "updated": "2008-11-23T17:42:13.000Z",
  "title": "The Mather measure and a Large Deviation Principle for the Entropy Penalized Method",
  "authors": [
    "Diogo A. Gomes",
    "Artur O. Lopes",
    "Joana Mohr"
  ],
  "categories": [
    "math.DS",
    "math.AP"
  ],
  "abstract": "We present a large deviation principle for the entropy penalized Mather problem when the Lagrangian L is generic (in this case the Mather measure $\\mu$ is unique and the support of $\\mu$ is the Aubry set). Consider, for each value of $\\epsilon $ and h, the entropy penalized Mather problem $\\min \\{\\int_{\\tn\\times\\rn} L(x,v)d\\mu(x,v)+\\epsilon S[\\mu]\\},$ where the entropy S is given by $S[\\mu]=\\int_{\\tn\\times\\rn}\\mu(x,v)\\ln\\frac{\\mu(x,v)}{\\int_{\\rn}\\mu(x,w)dw}dxdv,$ and the minimization is performed over the space of probability densities $\\mu(x,v)$ that satisfy the holonomy constraint It follows from D. Gomes and E. Valdinoci that there exists a minimizing measure $\\mu_{\\epsilon, h}$ which converges to the Mather measure $\\mu$. We show a LDP $\\lim_{\\epsilon,h\\to0} \\epsilon \\ln \\mu_{\\epsilon,h}(A),$ where $A\\subset \\mathbb{T}^N\\times\\mathbb{R}^N$. The deviation function I is given by $I(x,v)= L(x,v)+\\nabla\\phi_0(x)(v)-\\bar{H}_{0},$ where $\\phi_0$ is the unique viscosity solution for L.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-11-23T17:42:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A50",
      "37J50"
    ],
    "keywords": [
      "large deviation principle",
      "mather measure",
      "entropy penalized method",
      "entropy penalized mather problem",
      "unique viscosity solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0707.2603G"
    }
  }
}