{
  "id": "2001.12008",
  "version": "v1",
  "published": "2020-01-31T18:58:38.000Z",
  "updated": "2020-01-31T18:58:38.000Z",
  "title": "Conformal Skorokhod Embeddings of the uniform distribution and related extremal problems",
  "authors": [
    "Phanuel Mariano",
    "Hugo Panzo"
  ],
  "comment": "11 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\mu$ be a probability distribution with zero mean and finite variance. The conformal Skorokhod embedding problem (CSEP) asks for a simply connected domain $D\\subset\\mathbb{C}$ such that the real part of standard complex Brownian motion at its first exit time from $D$ has distribution $\\mu$ with the exit time having finite mean. The CSEP was first posed and solved by Gross (2019) where the author gives an explicit construction of a solution. In this paper we give an example of a simply connected domain $\\mathbb{U}$ that embeds the uniform distribution on $[-1,1]$ which differs from Gross' solution and possesses a certain extremal property. More precisely, if $D$ is any solution of the CSEP for the uniform distribution on $[-1,1]$, then the principal Dirichlet eigenvalue of $D$ is at least that of $\\mathbb{U}$. We also give general upper and lower bounds on the principal Dirichlet eigenvalue of a solution to the CSEP for any $\\mu$. The proofs rely on a spectral upper bound of the torsion function as well as a precise relation between the widths of the orthogonal projections of a simply connected domain and the support of its harmonic measure which is developed in the paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-31T18:58:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "uniform distribution",
      "related extremal problems",
      "simply connected domain",
      "principal dirichlet eigenvalue",
      "exit time"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}