{
  "id": "1412.3975",
  "version": "v1",
  "published": "2014-12-12T12:46:58.000Z",
  "updated": "2014-12-12T12:46:58.000Z",
  "title": "Construction and analysis of sticky reflected diffusions",
  "authors": [
    "Martin Grothaus",
    "Robert Voßhall"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We give a Dirichlet form approach for the construction of distorted Brownian motion in a bounded domain $\\Omega$ of $\\mathbb{R}^d$, $d \\geq 1$, with boundary $\\Gamma$, where the behavior at the boundary is sticky. The construction covers the case of a static boundary behavior as well as the case of a diffusion on the hypersurface $\\Gamma$ (for $d \\geq 2)$. More precisely, we consider the state space $\\overline{\\Omega}=\\Omega \\stackrel{.}{\\cup} \\Gamma$, the process is a diffusion process inside $\\Omega$, the occupation time of the process on the boundary $\\Gamma$ is positive and the process may diffuse on $\\Gamma$ as long as it sticks on the boundary. The problem is formulated in an $L^2$-setting and the construction is formulated under weak assumptions on the coefficients and $\\Gamma$. In order to analyze the process we assume a $C^2$-boundary and some weak differentiability conditions. In this case, we deduce that the process is also a solution to a given SDE for quasi every starting point in $\\overline{\\Omega}$ with respect to the underyling Dirichlet form. Under the addtional condition that $\\{ \\varrho =0 \\}$ is of capacity zero, we prove ergodicity of the constructed process and consequently, we verify that the boundary behavior is indeed sticky. Moreover, we show ($\\mathcal{L}^p$-)strong Feller properties which allow to characterize the constructed process even for every starting point in $\\overline{\\Omega} \\backslash \\{ \\varrho=0\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-12T12:46:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J50",
      "60J60",
      "58J65",
      "31C25",
      "35J25"
    ],
    "keywords": [
      "sticky reflected diffusions",
      "dirichlet form approach",
      "constructed process",
      "starting point",
      "strong feller properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.3975G"
    }
  }
}