{
  "id": "1402.0809",
  "version": "v2",
  "published": "2014-02-04T17:50:56.000Z",
  "updated": "2014-02-23T09:38:23.000Z",
  "title": "Large Deviations for stationary probabilities of a family of continuous time Markov chains via Aubry-Mather theory",
  "authors": [
    "Artur O. Lopes",
    "Adriana Neumann"
  ],
  "categories": [
    "math.DS",
    "cond-mat.stat-mech",
    "math.PR"
  ],
  "abstract": "We consider a family of continuous time symmetric random walks indexed by $k\\in \\mathbb{N}$, $\\{X_k(t),\\,t\\geq 0\\}$. For each $k\\in \\mathbb{N}$ the matching random walk take values in the finite set of states $\\Gamma_k=\\frac{1}{k}(\\mathbb{Z}/k\\mathbb{Z})$ which is a subset of the unitary circle. The stationary probability for such process converges to the uniform distribution on the circle, when $k\\to \\infty$. We disturb the system considering a fixed $C^2$ potential $V: \\mathbb{S}^1 \\to \\mathbb{R}$ and we will denote by $V_k$ the restriction of $V$ to $\\Gamma_k$. Then, we define a non-stochastic semigroup generated by the matrix $k\\,\\, L_k + k\\,\\, V_k$, where $k\\,\\, L_k $ is the infinifesimal generator of $\\{X_k(t),\\,t\\geq 0\\}$. From the continuous time Perron's Theorem one can normalized such semigroup, and, then we get another stochastic semigroup which generates a continuous time Markov Chain taking values on $\\Gamma_k$. The stationary probability vector for such Markov Chain is denoted by $\\pi_{k,V}$. We assume that the maximum of $V$ is attained in a unique point $x_0$ of $\\mathbb{S}^1$, and from this will follow that $\\pi_{k,V}\\to \\delta_{x_0}$. Our main goal is to analyze the large deviation principle for the family $\\pi_{k,V}$, when $k \\to\\infty$. The deviation function $I^V$, which is defined on $ \\mathbb{S}^1$, will be obtained from a procedure based on fixed points of the Lax-Oleinik operator and Aubry-Mather theory.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-02-23T09:38:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A50",
      "60F10",
      "37J50"
    ],
    "keywords": [
      "continuous time markov chain",
      "stationary probability",
      "large deviation",
      "aubry-mather theory",
      "time symmetric random walks"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s10955-015-1205-1",
      "journal": "Journal of Statistical Physics",
      "year": 2015,
      "month": "May",
      "pages": 797,
      "volume": 159,
      "number": 4
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015JSP...159..797L"
    }
  }
}