{
  "id": "2101.11999",
  "version": "v1",
  "published": "2021-01-28T13:54:07.000Z",
  "updated": "2021-01-28T13:54:07.000Z",
  "title": "Quasi-stationary distribution for the Langevin process in cylindrical domains, part I: existence, uniqueness and long-time convergence",
  "authors": [
    "Tony Lelièvre",
    "Mouad Ramil",
    "Julien Reygner"
  ],
  "categories": [
    "math.PR",
    "math.SP"
  ],
  "abstract": "Consider the Langevin process, described by a vector (position,momentum) in $\\mathbb{R}^{d}\\times\\mathbb{R}^d$. Let $\\mathcal O$ be a $\\mathcal{C}^2$ open bounded and connected set of $\\mathbb{R}^d$. We prove the compactness of the semigroup of the Langevin process absorbed at the boundary of the domain $D:=\\mathcal{O}\\times\\mathbb{R}^d$. We then obtain the existence of a unique quasi-stationary distribution (QSD) for the Langevin process on $D$. We also provide a spectral interpretation of this QSD and obtain an exponential convergence of the Langevin process conditioned on non-absorption towards the QSD.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-28T13:54:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "langevin process",
      "long-time convergence",
      "cylindrical domains",
      "uniqueness",
      "unique quasi-stationary distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}