{
  "id": "1604.06420",
  "version": "v1",
  "published": "2016-04-21T18:57:47.000Z",
  "updated": "2016-04-21T18:57:47.000Z",
  "title": "A Laplace Principle for Hermitian Brownian Motion and Free Entropy",
  "authors": [
    "Yoann Dabrowski"
  ],
  "comment": "80 pages, no figures. Comments Welcome!",
  "categories": [
    "math.PR",
    "math.OA"
  ],
  "abstract": "We prove that the $\\limsup$ and $\\liminf$ variants of Voiculescu's free entropy coincide. This is based on a Laplace principle (implying a full large deviation principle) for hermitian brownian motion on $[0,1]$. As a consequence, we show that microstates free entropy $\\chi(X_1,...,X_m)$ and non-microstate free entropy $\\chi^*(X_1,...,X_m)$ coincide for self-adjoint variables $(X_1,...,X_m)$ satisfying a Schwinger-Dyson equation for subquadratic, bounded bellow, strictly convex potentials with Lipschitz derivative sufficiently approximable by non-commutative polynomials. Applying the contraction principle, we obtain a large deviation result for Haar unitaries and deduce the most general additivity property for a new extended definition of orbital free entropy. Our results are based on Dupuis-Ellis weak convergence approach to large deviations, where one shows a Laplace principle in obtaining a stochastic control formulation for exponential functionals. In the non-commutative context, ultrapoduct analysis replaces weak-convergence of the stochastic control problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-21T18:57:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L54",
      "60F10",
      "60B20",
      "60G15",
      "46M07"
    ],
    "keywords": [
      "hermitian brownian motion",
      "laplace principle",
      "ultrapoduct analysis replaces weak-convergence",
      "full large deviation principle",
      "dupuis-ellis weak convergence approach"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 80,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160406420D"
    }
  }
}